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?

Relevance

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

• ballou
Lv 4
3 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