How can you find the inverse of a matrix A when given A and AB?

[fk378]

General question:
How do you determine a matrix B when given A and AB?

 

[steelphantom]

If you know the inverse of A, you can multiply on the left side by A^-1 to get B, i.e. (A^-1)AB = B.

 

[fk378]

Oh I see. I understand the computation, but could you explain the theory behind this?

 

[steelphantom]

This isn't exactly a thorough explanation, but basically if you multiply a matrix by its inverse, you get the identity matrix I. If you multiply any matrix by the identity matrix, you get the original matrix back again, e.g. IA = AI = A.

 

[HallsofIvy]

The theory? A guess it's things like "associative law" and "existence of the multiplicative identity"!
If you were given two numbers a, c, a not equal to 0, and told that ab= c, how would you solve for b?

If A has an inverse, then A^-1(AB)= (A^-1A)B= IB= B.

Notice the condition "If A has an inverse". There are plenty of different matrices, A, B, not having inverses, such that AB= 0. If A does not have an inverse, there may be many different matrices B such that AB= C.