# Math Help - Matrix/Scalar Multiplication

1. ## Matrix/Scalar Multiplication

Hey all, sorry for the vague title, I wasn't sure how to explain this.

I know that matrix multiplication is not commutative, but matrix scalar multiplication is (that is cA = Ac if c is a scalar and A is a matrix), so combining the two confused me.

So is this valid:

A is an n x n, invertible matrix. $\vec v$ is a vector with n elements, and c is a scalar.

$A^{-1}*c*A*\vec v = c*(A^{-1}*A)*\vec v$

My argument is that we are moving the scalar to the otherside of the matrix, not moving the matrix. And because rebracketing is fine for both Matrix multiplication and scalar multiplication, that should be fine.

So is the above equality alright?

Thanks,
-Keilan

2. Yes, this is correct.

$A^{-1}cA = (A^{-1}c)A = (cA^{-1})A = c(A^{-1}A) = cI$