# Proving the Associative Property of Matrix Multiplication?

I know that multiplication of matrices is associative. However, I don't know what I'm doing wrong.

Assume

A= { a , b }

{ c , d }

B= { e , f }

{ g , h }

C= { i , j }

{ k , l }

If (AB)C = A(BC), then matrix multiplication is associative.

AB = { ae+bg , af+ bh}

{ ce+dg , cf+dh }

(AB)C = { aei+bgi+afk+bhk , aej+bgj+afl+bhl }

{ cei+dgi+cfk+dhk , cej+dgj+cfl+dhl }

BC = { ei+fk , ej+fl }

{ gi+hk , gj+hl }

A(BC) = { aei+fka+ejc+flc , eib+fkb+ejc+flc }

{ gia+hka+gjc+hlc , gib+hkb+gjc+hlc}

If you go through, you see that the variables don't match up.

Why are these variables not matching up?

### 2 Answers

- Demiurge42Lv 71 decade agoFavorite Answer
You did A(BC) wrong

It looks like you did (BC)A instead

Left multiplication and right multiplication are not the same thing when using matrices

- ballouLv 43 years ago
If each and each of A, B, and C is invertible, then det(A) ? 0 det(B) ? 0, and det(C) ? 0. ABC invertible <=> det(ABC) ? 0 that's clearly genuine on condition that det(ABC) = det(A)*det(B)*det(C) = (fabricated from nonzero components) ? 0