What is the Proof for Matrix Multiplication with Invertible Matrices?

[jumbogala]

Homework Statement

I'm doing a proof in which I need to show:
given that AX = 0, AVX=0 where V is invertible.

Also, given that AVY = 0, then AY = 0.

Homework Equations

The Attempt at a Solution

I can't remember from the previous course I took how to do this. I know that I can multiply from the left or right by V^-1, but seeing as V is in the middle that won't work.

This is part of a larger proof, if it would make more sense to have the entire question let me know.

 

[rasmhop]

I'm assuming A,X,V,Y are matrices, but I'm not sure (EDIT: I see you stated in the title that they are matrices). Also do X and Y need to be vectors or are they general matrices. Do we require some matrices to be square or non-zero? Any other assumptions?

The information you have given is not sufficient. Consider:
$$A = \left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right]$$
$$X = \left[\begin{array}{cc} 0 \\ 0 \end{array} \right]$$
$$V = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]$$
$$Y = \left[\begin{array}{cc} 1 \\ 0 \end{array} \right]$$
Then $V^2=I$ so V is invertible. AX = AVX = AVY = 0, but,
$$AY = \left[\begin{array}{cc} 1 \\ 0 \end{array} \right]$$

 

[rasmhop]

A slightly more interesting example where all matrices are non-zero:
$$A = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 &0 \\ 0 &0 & 0 \end{array} \right]$$
$$X = \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right]$$
$$V = \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1&0&0 \end{array} \right]$$
$$Y = \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right]$$
Then $V^3=I$ so V is invertible. AX = AVX = AVY = 0, but,
$$AY = \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right]$$