In matrix multiplication, does ABC=ACB if A, B, and C are nxn matrices?

1 Answer
Oct 24, 2015

In general, no.

Matrix multiplication is associative but not commutative.

Explanation:

Let A=( (1, 0), (0, 1) ), B=((1, 1), (0, 0)) and C=((1, 0), (1, 0))

Then:

ABC = BC = ((1, 1), (0, 0))((1, 0), (1, 0)) = ((2, 0), (0, 0))

ACB = CB = ((1, 0), (1, 0))((1, 1), (0, 0)) = ((1, 1), (1, 1))

What is true for all n xx n matrices A, B and C is:

(AB)C = A(BC)

That is, matrix multiplication is associative.