# Matrix Theory (Matrix Inverse test question)

1. Aug 7, 2012

### mateomy

I was asked this question on a test I just took and I was kinda stumped. I worked it out but I think I did it incorrectly so I was hoping for some input. Thanks.

'A' is a 3x3 matrix and 'I' is the identity matrix for a 3x3. Find the Inverse of 'A' given:

$$(\mathbf{A}\,+\,\mathbf{I})^{2}\,=\,\mathbf{0}$$

Where '0' is the zero matrix.

So what I did (probably ignorantly) was treat it like any standard algebraic equation; took the radical of both sides, and subtracted the Identity from each side. So I had the zero matrix minus the standard identity 3x3. So I ended up getting a negative identity matrix on the right hand side. So I can see that 'A' equals the negative of an identity matrix. So to find the identity of that I put my 'A' next to and identity to solve in the standard fashion, for the identity of A. Which just happened to be, again, a negative of a 3x3 identity matrix.

Sorry I didn't feel like Latex'ing out all the matrix work but I think it can be followed by what I wrote.

Thanks again.

2. Aug 7, 2012

### Ray Vickson

If B is a matrix and $B^2 = 0,$ that does NOT imply that B must be zero; matrices do not behave like real numbers in all respects. However, you _can_ use standard algebra to expand $(A+I)^2$, because the matrices A and I commute under multiplication (that is, AI = IA), and you _can_ use AB+AC = A(B+C), etc.

RGV