Step-by-step derivation
S001 · implies
Before
source roles = [qf-determinant-scalar, scaled_determinant, domain, reals]
After
derived = [[integer_det_cM]]
Conditions: none
Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.
Step identifiers and dependencies
Rule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001
S002 · calculation
Before
Det scalar ORIGINAL entry0,0 := mul(-4, 7)
After
Det scalar ORIGINAL entry0,0 = -28
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Det scalar ORIGINAL entry0,1 := mul(-4, 4)
After
Det scalar ORIGINAL entry0,1 = -16
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Det scalar ORIGINAL entry1,0 := mul(-4, -2)
After
Det scalar ORIGINAL entry1,0 = 8
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Det scalar ORIGINAL entry1,1 := mul(-4, 20)
After
Det scalar ORIGINAL entry1,1 = -80
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [[[7, 4], [-2, 20]], -4]
After
derived = [[[-28, -16], [8, -80]]]
Conditions: none
Scale ALL FOUR entries BEFORE taking the determinant.TWO row factors in dimension2 give c²det(M),NOT c det(M).Zero/negative c is legal.
Step identifiers and dependencies
Rule: Determinant_SCALE_ALL_entries_c2_factor_in_dimension2 · Uses: G001, S005
S007 · calculation
Before
Det requested matrix ORIGINAL main diagonal product := mul(-28, -80)
After
Det requested matrix ORIGINAL main diagonal product = 2240
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Det requested matrix ORIGINAL other diagonal product := mul(-16, 8)
After
Det requested matrix ORIGINAL other diagonal product = -128
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Det requested matrix ORIGINAL signed ad minus bc := sub(2240, -128)
After
Det requested matrix ORIGINAL signed ad minus bc = 2368
Conditions: none
Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [Det requested matrix, [[-28, -16], [8, -80]]]
After
derived = [2240, -128, 2368]
Conditions: none
For THIS2×2 matrix explicitly calculate ad and bc,then retain the sign in ad−bc,not the permanent ad+bc.Derived entries are not SAFE-recapped.
Step identifiers and dependencies
Rule: Determinant_2x2_signed_ad_MINUS_bc · Uses: G001, S009