# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Matrices: dimensions, one-based indexing, entrywise addition and transpose](/en/derivations/derivation-mko-matrix-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-matrix)

Let M=[[2,−1,0],[5,3,7]] and N=[[1,4,−2],[0,−3,6]] (each inner bracket is one row). Find their dimensions, read off M(1,2) and M(2,3), and decide whether M(3,1) exists; compute M+N, Mᵀ and Nᵀ; prove (A+B)ᵀ=Aᵀ+Bᵀ and (Aᵀ)ᵀ=A for all real matrices of equal size and check them on M and N; explain why M+Mᵀ is undefined; finally use counterexamples to separate the transpose from reversing the rows and from the inverse matrix, and translate zero-based program indices back to one-based indices.

Finite matrices: dimensions, indexing, and transpose

derivation-mko-matrix-01



Concept guide



## [Full RREF and rank bounds](/en/derivations/derivation-mko-matrix-rank-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-matrix)

A=[[1,2,3],[2,4,6],[1,1,1]] has row and column rank 2.

Matrix rank

derivation-mko-matrix-rank-01





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