Can You Find Matrices B and C Where AB = AC but B ≠ C?

[LaraCroft]

Let A =

[ -3 9 ]
[ -1 3 ]

Find two 2x2 matrices B and C, such that AB = AC BUT B does not equal C.

I am stumped, but I am trying inverses and such, with no luck :(

Any insight would be great!

 

[John Creighto]

Let A =

[ -3 9 ]
[ -1 3 ]

Find two 2x2 matrices B and C, such that AB = AC BUT B does not equal C.

I am stumped, but I am trying inverses and such, with no luck :(

Any insight would be great!

Matrix A has rank 1. Any component of the column vectors in B and C in the null space of A will be mapped to zero. This leaves quite a few possibilities.

However, if you haven't learned about null spaces yet, Just use variables for the ellements of the B matrix, multiply A by B and see how these variables can be changed without affecting the product.

This might be more clear if you row reduce A. Remember that each row operation can be represented by a linear transformation. Doing the row operation on the identity matrix will give you the desired transformation. Therefor you can safely row reduce A without affecting the solution to the problem.