Matrix Determinant • det(AB) = det(A)·det(B)

Problem

Show det(AB) equals det(A) times det(B) for square matrices A, B

We illustrate with 2×2 and 3×3 integer matrices and multiple independent checks.

Answer

True. For all square matrices A, B of the same size, det(AB) = det(A)·det(B).

Reason Why

Mathematical English justification (elementary matrices argument)

  1. Consider the three elementary row operations, implemented by left-multiplying with an elementary matrix E:
    • Swap two rows: det(E) = −1 and det(EA) = −det(A).
    • Scale row i by λ: det(E) = λ and det(EA) = λ·det(A).
    • Add λ times row j to row i (i ≠ j): det(E) = 1 and det(EA) = det(A).
    In each case, det(EA) = det(E)·det(A).
  2. Every invertible matrix A can be written as a product of elementary matrices: A = E_k … E_1. Then det(AB) = det(E_k … E_1 B) = det(E_k) … det(E_1) det(B) = det(A) det(B).
  3. If A is singular, then AB is singular, so det(AB) = 0 = det(A) det(B). Therefore the identity holds for all A, B.
Check (harness) #1

Fixed example (3×3)

Compute det(AB) and compare with det(A)·det(B).

Check (harness) #2

Random 3×3 integer matrices (200 trials)

Entries from −3..3. We use the exact 3×3 formula (no floating error).

Check (harness) #3

Upper-triangular matrices

For triangular U, V: det(UV) equals product of diagonals of U times product of diagonals of V.

Check (harness) #4

Elementary matrices

Pick a random A and a random elementary E. Verify det(EA) = det(E)·det(A) for each type.

Check (harness) #5

Singular case

If det(A) = 0, then det(AB) must be 0 for any B.

Check (harness) #6

Your matrices (CSV)

Enter 2×2 or 3×3 matrices as rows separated by semicolons. Examples:
A: 1,2; 3,4
B: 2,0; 5,1 or
A: 1,0,2; -1,3,1; 0,2,1

Check (harness) #7

Power rule (consequence)

det(A^k) = det(A)^k for k = 2,3,4 (follows from multiplicativity).