# Finite Real Matrix Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261006-advance-20261007-r01-matrix-foundations · 2026-10-06 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Dimensions of a Nonempty Matrix

The real matrix M=[[-12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(1, 1)

Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 1). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0001/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 2. Dimensions of a Nonempty Matrix

The real matrix M=[[-11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(1, 2)

Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 2). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0002/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 3. Dimensions of a Nonempty Matrix

The real matrix M=[[-10, -11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(1, 3)

Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 3). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0003/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 4. Dimensions of a Nonempty Matrix

The real matrix M=[[-9, -10, -11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(1, 4)

Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 4). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0004/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 5. Dimensions of a Nonempty Matrix

The real matrix M=[[-8, -9, -10, -11, -12]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(1, 5)

Count the nonempty equal-length rows and entries per row to obtain dimensions (1, 5). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0005/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 6. Dimensions of a Nonempty Matrix

The real matrix M=[[-7], [-5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(2, 1)

Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 1). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0006/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 7. Dimensions of a Nonempty Matrix

The real matrix M=[[-6, -7], [-4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(2, 2)

Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 2). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0007/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 8. Dimensions of a Nonempty Matrix

The real matrix M=[[-5, -6, -7], [-3, -4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(2, 3)

Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 3). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0008/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 9. Dimensions of a Nonempty Matrix

The real matrix M=[[-4, -5, -6, -7], [-2, -3, -4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(2, 4)

Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 4). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0009/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 10. Dimensions of a Nonempty Matrix

The real matrix M=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(2, 5)

Count the nonempty equal-length rows and entries per row to obtain dimensions (2, 5). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0010/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 11. Dimensions of a Nonempty Matrix

The real matrix M=[[-2], [0], [2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(3, 1)

Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 1). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0011/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 12. Dimensions of a Nonempty Matrix

The real matrix M=[[-1, -2], [1, 0], [3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(3, 2)

Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 2). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0012/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 13. Dimensions of a Nonempty Matrix

The real matrix M=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(3, 3)

Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 3). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0013/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 14. Dimensions of a Nonempty Matrix

The real matrix M=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(3, 4)

Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 4). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0014/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 15. Dimensions of a Nonempty Matrix

The real matrix M=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(3, 5)

Count the nonempty equal-length rows and entries per row to obtain dimensions (3, 5). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0015/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 16. Dimensions of a Nonempty Matrix

The real matrix M=[[3], [5], [7], [9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(4, 1)

Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 1). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0016/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 17. Dimensions of a Nonempty Matrix

The real matrix M=[[4, 3], [6, 5], [8, 7], [10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(4, 2)

Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 2). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0017/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 18. Dimensions of a Nonempty Matrix

The real matrix M=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(4, 3)

Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 3). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0018/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 19. Dimensions of a Nonempty Matrix

The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(4, 4)

Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 4). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0019/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 20. Dimensions of a Nonempty Matrix

The real matrix M=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(4, 5)

Count the nonempty equal-length rows and entries per row to obtain dimensions (4, 5). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0020/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 21. Dimensions of a Nonempty Matrix

The real matrix M=[[8], [10], [12], [14], [16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(5, 1)

Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 1). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0021/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 22. Dimensions of a Nonempty Matrix

The real matrix M=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(5, 2)

Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 2). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0022/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 23. Dimensions of a Nonempty Matrix

The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(5, 3)

Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 3). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0023/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 24. Dimensions of a Nonempty Matrix

The real matrix M=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(5, 4)

Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 4). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0024/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 25. Dimensions of a Nonempty Matrix

The real matrix M=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] is displayed as nested rows. Each dimension is from one through five. Give (number of rows, number of columns).

View answer and explanation(5, 5)

Count the nonempty equal-length rows and entries per row to obtain dimensions (5, 5). The total number of entries is not either dimension by itself.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0025/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-shape · candidate





## 26. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.

View answer and explanation-12

Read M(1,1)=-12 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0026/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 27. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.

View answer and explanation-11

Read M(1,1)=-11 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0027/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 28. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-10, -11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.

View answer and explanation-10

Read M(1,1)=-10 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0028/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 29. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-9, -10, -11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.

View answer and explanation-9

Read M(1,1)=-9 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0029/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 30. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-8, -9, -10, -11, -12]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.

View answer and explanation-8

Read M(1,1)=-8 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0030/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 31. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-7], [-5]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.

View answer and explanation-5

Read M(2,1)=-5 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0031/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 32. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-6, -7], [-4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 2.

View answer and explanation-7

Read M(1,2)=-7 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0032/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 33. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-5, -6, -7], [-3, -4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 2.

View answer and explanation-4

Read M(2,2)=-4 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0033/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 34. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-4, -5, -6, -7], [-2, -3, -4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 2.

View answer and explanation-5

Read M(1,2)=-5 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0034/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 35. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 2.

View answer and explanation-2

Read M(2,2)=-2 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0035/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 36. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-2], [0], [2]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.

View answer and explanation0

Read M(2,1)=0 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0036/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 37. Specified Matrix Entry with One-Based Indices

The real matrix M=[[-1, -2], [1, 0], [3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 1.

View answer and explanation3

Read M(3,1)=3 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0037/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 38. Specified Matrix Entry with One-Based Indices

The real matrix M=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 3.

View answer and explanation-2

Read M(1,3)=-2 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0038/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 39. Specified Matrix Entry with One-Based Indices

The real matrix M=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 3.

View answer and explanation1

Read M(2,3)=1 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0039/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 40. Specified Matrix Entry with One-Based Indices

The real matrix M=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 3.

View answer and explanation4

Read M(3,3)=4 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0040/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 41. Specified Matrix Entry with One-Based Indices

The real matrix M=[[3], [5], [7], [9]] is displayed as nested rows. Using one-based indices, find the entry in row 4, column 1.

View answer and explanation9

Read M(4,1)=9 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0041/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 42. Specified Matrix Entry with One-Based Indices

The real matrix M=[[4, 3], [6, 5], [8, 7], [10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 2.

View answer and explanation3

Read M(1,2)=3 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0042/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 43. Specified Matrix Entry with One-Based Indices

The real matrix M=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.

View answer and explanation7

Read M(2,1)=7 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0043/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 44. Specified Matrix Entry with One-Based Indices

The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 4.

View answer and explanation7

Read M(3,4)=7 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0044/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 45. Specified Matrix Entry with One-Based Indices

The real matrix M=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] is displayed as nested rows. Using one-based indices, find the entry in row 4, column 4.

View answer and explanation10

Read M(4,4)=10 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0045/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 46. Specified Matrix Entry with One-Based Indices

The real matrix M=[[8], [10], [12], [14], [16]] is displayed as nested rows. Using one-based indices, find the entry in row 1, column 1.

View answer and explanation8

Read M(1,1)=8 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0046/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 47. Specified Matrix Entry with One-Based Indices

The real matrix M=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 2, column 1.

View answer and explanation11

Read M(2,1)=11 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0047/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 48. Specified Matrix Entry with One-Based Indices

The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 3, column 2.

View answer and explanation13

Read M(3,2)=13 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0048/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 49. Specified Matrix Entry with One-Based Indices

The real matrix M=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 4, column 1.

View answer and explanation17

Read M(4,1)=17 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0049/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 50. Specified Matrix Entry with One-Based Indices

The real matrix M=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] is displayed as nested rows. Using one-based indices, find the entry in row 5, column 5.

View answer and explanation16

Read M(5,5)=16 under the declared one-based convention. Both indices are within the dimensions, and other entries do not change this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0050/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-entry · candidate





## 51. Addition of Equal-Size Matrices

Over R, A=[[-12]] and B=[[12]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0]]

Add entries in corresponding positions to obtain [[0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0051/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 52. Addition of Equal-Size Matrices

Over R, A=[[-11, -12]] and B=[[2, 4]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8]]

Add entries in corresponding positions to obtain [[-9, -8]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0052/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 53. Addition of Equal-Size Matrices

Over R, A=[[-10, -11, -12]] and B=[[1, 3, 5]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7]]

Add entries in corresponding positions to obtain [[-9, -8, -7]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0053/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 54. Addition of Equal-Size Matrices

Over R, A=[[-9, -10, -11, -12]] and B=[[9, 10, 11, 12]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0, 0, 0]]

Add entries in corresponding positions to obtain [[0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0054/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 55. Addition of Equal-Size Matrices

Over R, A=[[-8, -9, -10, -11, -12]] and B=[[-1, 1, 3, 5, 7]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7, -6, -5]]

Add entries in corresponding positions to obtain [[-9, -8, -7, -6, -5]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0055/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 56. Addition of Equal-Size Matrices

Over R, A=[[-7], [-5]] and B=[[-2], [-1]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9], [-6]]

Add entries in corresponding positions to obtain [[-9], [-6]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0056/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 57. Addition of Equal-Size Matrices

Over R, A=[[-6, -7], [-4, -5]] and B=[[6, 7], [4, 5]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0], [0, 0]]

Add entries in corresponding positions to obtain [[0, 0], [0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0057/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 58. Addition of Equal-Size Matrices

Over R, A=[[-5, -6, -7], [-3, -4, -5]] and B=[[-4, -2, 0], [-3, -1, 1]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7], [-6, -5, -4]]

Add entries in corresponding positions to obtain [[-9, -8, -7], [-6, -5, -4]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0058/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 59. Addition of Equal-Size Matrices

Over R, A=[[-4, -5, -6, -7], [-2, -3, -4, -5]] and B=[[-5, -3, -1, 1], [-4, -2, 0, 2]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7, -6], [-6, -5, -4, -3]]

Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0059/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 60. Addition of Equal-Size Matrices

Over R, A=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] and B=[[3, 4, 5, 6, 7], [1, 2, 3, 4, 5]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]

Add entries in corresponding positions to obtain [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0060/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 61. Addition of Equal-Size Matrices

Over R, A=[[-2], [0], [2]] and B=[[-7], [-6], [-5]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9], [-6], [-3]]

Add entries in corresponding positions to obtain [[-9], [-6], [-3]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0061/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 62. Addition of Equal-Size Matrices

Over R, A=[[-1, -2], [1, 0], [3, 2]] and B=[[-8, -6], [-7, -5], [-6, -4]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8], [-6, -5], [-3, -2]]

Add entries in corresponding positions to obtain [[-9, -8], [-6, -5], [-3, -2]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0062/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 63. Addition of Equal-Size Matrices

Over R, A=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] and B=[[0, 1, 2], [-2, -1, 0], [-4, -3, -2]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0, 0], [0, 0, 0], [0, 0, 0]]

Add entries in corresponding positions to obtain [[0, 0, 0], [0, 0, 0], [0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0063/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 64. Addition of Equal-Size Matrices

Over R, A=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] and B=[[-10, -8, -6, -4], [-9, -7, -5, -3], [-8, -6, -4, -2]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0]]

Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0064/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 65. Addition of Equal-Size Matrices

Over R, A=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] and B=[[-11, -9, -7, -5, -3], [-10, -8, -6, -4, -2], [-9, -7, -5, -3, -1]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1]]

Add entries in corresponding positions to obtain [[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0065/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 66. Addition of Equal-Size Matrices

Over R, A=[[3], [5], [7], [9]] and B=[[-3], [-5], [-7], [-9]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0], [0], [0], [0]]

Add entries in corresponding positions to obtain [[0], [0], [0], [0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0066/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 67. Addition of Equal-Size Matrices

Over R, A=[[4, 3], [6, 5], [8, 7], [10, 9]] and B=[[-13, -11], [-12, -10], [-11, -9], [-10, -8]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8], [-6, -5], [-3, -2], [0, 1]]

Add entries in corresponding positions to obtain [[-9, -8], [-6, -5], [-3, -2], [0, 1]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0067/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 68. Addition of Equal-Size Matrices

Over R, A=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] and B=[[-14, -12, -10], [-13, -11, -9], [-12, -10, -8], [-11, -9, -7]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2]]

Add entries in corresponding positions to obtain [[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0068/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 69. Addition of Equal-Size Matrices

Over R, A=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] and B=[[-6, -5, -4, -3], [-8, -7, -6, -5], [-10, -9, -8, -7], [-12, -11, -10, -9]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]

Add entries in corresponding positions to obtain [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0069/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 70. Addition of Equal-Size Matrices

Over R, A=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] and B=[[-16, -14, -12, -10, -8], [-15, -13, -11, -9, -7], [-14, -12, -10, -8, -6], [-13, -11, -9, -7, -5]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1], [0, 1, 2, 3, 4]]

Add entries in corresponding positions to obtain [[-9, -8, -7, -6, -5], [-6, -5, -4, -3, -2], [-3, -2, -1, 0, 1], [0, 1, 2, 3, 4]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0070/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 71. Addition of Equal-Size Matrices

Over R, A=[[8], [10], [12], [14], [16]] and B=[[-17], [-16], [-15], [-14], [-13]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9], [-6], [-3], [0], [3]]

Add entries in corresponding positions to obtain [[-9], [-6], [-3], [0], [3]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0071/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 72. Addition of Equal-Size Matrices

Over R, A=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] and B=[[-9, -8], [-11, -10], [-13, -12], [-15, -14], [-17, -16]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]

Add entries in corresponding positions to obtain [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0072/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 73. Addition of Equal-Size Matrices

Over R, A=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] and B=[[-19, -17, -15], [-18, -16, -14], [-17, -15, -13], [-16, -14, -12], [-15, -13, -11]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2], [3, 4, 5]]

Add entries in corresponding positions to obtain [[-9, -8, -7], [-6, -5, -4], [-3, -2, -1], [0, 1, 2], [3, 4, 5]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0073/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 74. Addition of Equal-Size Matrices

Over R, A=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] and B=[[-20, -18, -16, -14], [-19, -17, -15, -13], [-18, -16, -14, -12], [-17, -15, -13, -11], [-16, -14, -12, -10]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0], [0, 1, 2, 3], [3, 4, 5, 6]]

Add entries in corresponding positions to obtain [[-9, -8, -7, -6], [-6, -5, -4, -3], [-3, -2, -1, 0], [0, 1, 2, 3], [3, 4, 5, 6]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0074/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 75. Addition of Equal-Size Matrices

Over R, A=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] and B=[[-12, -11, -10, -9, -8], [-14, -13, -12, -11, -10], [-16, -15, -14, -13, -12], [-18, -17, -16, -15, -14], [-20, -19, -18, -17, -16]] have the same dimensions. Find A+B as a nested list of rows.

View answer and explanation[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]

Add entries in corresponding positions to obtain [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]. Do not pad matrices of different dimensions with zeros to force this operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0075/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-addition · candidate





## 76. Transpose of a Rectangular Matrix

The real matrix M=[[-12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-12]]

Move each entry from row i, column j to row j, column i, obtaining [[-12]]. The dimensions change from 1×1 to 1×1; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0076/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 77. Transpose of a Rectangular Matrix

The real matrix M=[[-11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-11], [-12]]

Move each entry from row i, column j to row j, column i, obtaining [[-11], [-12]]. The dimensions change from 1×2 to 2×1; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0077/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 78. Transpose of a Rectangular Matrix

The real matrix M=[[-10, -11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-10], [-11], [-12]]

Move each entry from row i, column j to row j, column i, obtaining [[-10], [-11], [-12]]. The dimensions change from 1×3 to 3×1; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0078/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 79. Transpose of a Rectangular Matrix

The real matrix M=[[-9, -10, -11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-9], [-10], [-11], [-12]]

Move each entry from row i, column j to row j, column i, obtaining [[-9], [-10], [-11], [-12]]. The dimensions change from 1×4 to 4×1; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0079/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 80. Transpose of a Rectangular Matrix

The real matrix M=[[-8, -9, -10, -11, -12]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-8], [-9], [-10], [-11], [-12]]

Move each entry from row i, column j to row j, column i, obtaining [[-8], [-9], [-10], [-11], [-12]]. The dimensions change from 1×5 to 5×1; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0080/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 81. Transpose of a Rectangular Matrix

The real matrix M=[[-7], [-5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-7, -5]]

Move each entry from row i, column j to row j, column i, obtaining [[-7, -5]]. The dimensions change from 2×1 to 1×2; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0081/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 82. Transpose of a Rectangular Matrix

The real matrix M=[[-6, -7], [-4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-6, -4], [-7, -5]]

Move each entry from row i, column j to row j, column i, obtaining [[-6, -4], [-7, -5]]. The dimensions change from 2×2 to 2×2; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0082/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 83. Transpose of a Rectangular Matrix

The real matrix M=[[-5, -6, -7], [-3, -4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-5, -3], [-6, -4], [-7, -5]]

Move each entry from row i, column j to row j, column i, obtaining [[-5, -3], [-6, -4], [-7, -5]]. The dimensions change from 2×3 to 3×2; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0083/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 84. Transpose of a Rectangular Matrix

The real matrix M=[[-4, -5, -6, -7], [-2, -3, -4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-4, -2], [-5, -3], [-6, -4], [-7, -5]]

Move each entry from row i, column j to row j, column i, obtaining [[-4, -2], [-5, -3], [-6, -4], [-7, -5]]. The dimensions change from 2×4 to 4×2; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0084/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 85. Transpose of a Rectangular Matrix

The real matrix M=[[-3, -4, -5, -6, -7], [-1, -2, -3, -4, -5]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-3, -1], [-4, -2], [-5, -3], [-6, -4], [-7, -5]]

Move each entry from row i, column j to row j, column i, obtaining [[-3, -1], [-4, -2], [-5, -3], [-6, -4], [-7, -5]]. The dimensions change from 2×5 to 5×2; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0085/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 86. Transpose of a Rectangular Matrix

The real matrix M=[[-2], [0], [2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-2, 0, 2]]

Move each entry from row i, column j to row j, column i, obtaining [[-2, 0, 2]]. The dimensions change from 3×1 to 1×3; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0086/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 87. Transpose of a Rectangular Matrix

The real matrix M=[[-1, -2], [1, 0], [3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[-1, 1, 3], [-2, 0, 2]]

Move each entry from row i, column j to row j, column i, obtaining [[-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×2 to 2×3; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0087/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 88. Transpose of a Rectangular Matrix

The real matrix M=[[0, -1, -2], [2, 1, 0], [4, 3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[0, 2, 4], [-1, 1, 3], [-2, 0, 2]]

Move each entry from row i, column j to row j, column i, obtaining [[0, 2, 4], [-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×3 to 3×3; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0088/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 89. Transpose of a Rectangular Matrix

The real matrix M=[[1, 0, -1, -2], [3, 2, 1, 0], [5, 4, 3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]

Move each entry from row i, column j to row j, column i, obtaining [[1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×4 to 4×3; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0089/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 90. Transpose of a Rectangular Matrix

The real matrix M=[[2, 1, 0, -1, -2], [4, 3, 2, 1, 0], [6, 5, 4, 3, 2]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[2, 4, 6], [1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]

Move each entry from row i, column j to row j, column i, obtaining [[2, 4, 6], [1, 3, 5], [0, 2, 4], [-1, 1, 3], [-2, 0, 2]]. The dimensions change from 3×5 to 5×3; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0090/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 91. Transpose of a Rectangular Matrix

The real matrix M=[[3], [5], [7], [9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[3, 5, 7, 9]]

Move each entry from row i, column j to row j, column i, obtaining [[3, 5, 7, 9]]. The dimensions change from 4×1 to 1×4; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0091/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 92. Transpose of a Rectangular Matrix

The real matrix M=[[4, 3], [6, 5], [8, 7], [10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[4, 6, 8, 10], [3, 5, 7, 9]]

Move each entry from row i, column j to row j, column i, obtaining [[4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×2 to 2×4; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0092/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 93. Transpose of a Rectangular Matrix

The real matrix M=[[5, 4, 3], [7, 6, 5], [9, 8, 7], [11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]

Move each entry from row i, column j to row j, column i, obtaining [[5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×3 to 3×4; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0093/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 94. Transpose of a Rectangular Matrix

The real matrix M=[[6, 5, 4, 3], [8, 7, 6, 5], [10, 9, 8, 7], [12, 11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]

Move each entry from row i, column j to row j, column i, obtaining [[6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×4 to 4×4; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0094/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 95. Transpose of a Rectangular Matrix

The real matrix M=[[7, 6, 5, 4, 3], [9, 8, 7, 6, 5], [11, 10, 9, 8, 7], [13, 12, 11, 10, 9]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[7, 9, 11, 13], [6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]

Move each entry from row i, column j to row j, column i, obtaining [[7, 9, 11, 13], [6, 8, 10, 12], [5, 7, 9, 11], [4, 6, 8, 10], [3, 5, 7, 9]]. The dimensions change from 4×5 to 5×4; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0095/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 96. Transpose of a Rectangular Matrix

The real matrix M=[[8], [10], [12], [14], [16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[8, 10, 12, 14, 16]]

Move each entry from row i, column j to row j, column i, obtaining [[8, 10, 12, 14, 16]]. The dimensions change from 5×1 to 1×5; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0096/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 97. Transpose of a Rectangular Matrix

The real matrix M=[[9, 8], [11, 10], [13, 12], [15, 14], [17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]

Move each entry from row i, column j to row j, column i, obtaining [[9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×2 to 2×5; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0097/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 98. Transpose of a Rectangular Matrix

The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]

Move each entry from row i, column j to row j, column i, obtaining [[10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×3 to 3×5; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0098/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 99. Transpose of a Rectangular Matrix

The real matrix M=[[11, 10, 9, 8], [13, 12, 11, 10], [15, 14, 13, 12], [17, 16, 15, 14], [19, 18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]

Move each entry from row i, column j to row j, column i, obtaining [[11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×4 to 4×5; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0099/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate





## 100. Transpose of a Rectangular Matrix

The real matrix M=[[12, 11, 10, 9, 8], [14, 13, 12, 11, 10], [16, 15, 14, 13, 12], [18, 17, 16, 15, 14], [20, 19, 18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows.

View answer and explanation[[12, 14, 16, 18, 20], [11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]

Move each entry from row i, column j to row j, column i, obtaining [[12, 14, 16, 18, 20], [11, 13, 15, 17, 19], [10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]]. The dimensions change from 5×5 to 5×5; this is not row reversal or matrix inversion.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r01-matrix-0100/)

Finite matrices: dimensions, indexing, and transpose · qf-matrix-transpose · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r01-matrix-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261006-advance-20261007-r01-matrix-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r01-matrix-foundations/validation-report.json)