Homework Help: Square matrix and its transpose satisfying an equation

1. Mar 25, 2015

Lord Anoobis

1. The problem statement, all variables and given/known data
Show that if a square matrix A satisfies

A3 + 4A2 -2A + 7I = 0
Mod note: It took me a little while to realize that the last term on the left is 7I, seven times the identity matrix. The italicized I character without serifs appeared to me to be the slash character /.

then so does AT

2. Relevant equations

3. The attempt at a solution
What I notice is that for any n x n matrix A and powers thereof, the diagonals of A and the transpose are the same. I experimented with a 2 x 2 matrix (with entries a, b, c, d), squared and cubed to see what happens and the result, aside from being somewhat messy, ends with each matrix reducing to I when appropriate row operations are applied. I'm not sure how to proceed from here or if I'm even on the right track with this thinking. Please assist.

Last edited by a moderator: Mar 25, 2015
2. Mar 25, 2015

Dick

You are thinking about this too hard. Just take the transpose of both sides of that equation. For example, what is $(A^3)^T$ in terms of $A^T$.

3. Mar 25, 2015

Lord Anoobis

Damn. I really should have seen that, unbelievably simple. Of course , hindsight is always 20/20. Thanks.