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

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To find matrices B and C such that AB = AC but B ≠ C, it is essential to understand that matrix A has rank 1, meaning its null space allows for multiple solutions. Any components of B and C that lie in the null space of A will result in the same product when multiplied by A, leading to the desired equality. Using variables for the elements of matrix B and performing row operations on A can help identify how these variables can change without affecting the product. Row reducing A while applying the same transformations to the identity matrix can clarify the relationships between B and C. This approach provides a systematic way to explore potential matrices that meet the criteria.
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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!:wink:
 
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LaraCroft said:
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!:wink:

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.
 

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