Question

How do you find the inverse of `A=``((3, 2, 7), (0, 1, 2))`?

Answer 1

The additive inverse of `A = ((3, 2, 7), (0, 1, 2))` is `-A = ((-3, -2, -7), (0, -1, -2))`

It has no multiplicative inverse. Only square matrices have those.

Explanation:

`((3, 2, 7),(0, 1, 2)) + ((-3, -2, -7), (0, -1, -2)) = ((0, 0, 0), (0, 0, 0))`

So `A = ((3, 2, 7), (0, 1, 2))` has an additive inverse `-A = ((-3, -2, -7), (0, -1, -2))`

A multiplicative inverse would have to satisfy:

`A A^(-1) = A^(-1) A = I`

for some identity matrix `I`. Would `I` be a `2 xx 2` matrix or a `3 xx 3` one?

By considering how we multiply matrices, then we notice that `AB` and `BA` can only be square matrices if `B` is a `3 xx 2` matrix. If `B` is a `3 xx 2` matrix then `AB` is `2xx2` and `BA` is `3xx3`.