Square matrix and its transpose satisfying an equation

[Lord Anoobis]

Homework Statement

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 A^T

Homework Equations

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.

 

[Dick]

Homework Statement

Show that if a square matrix A satisfies

A³ + 4A² -2A + 7I = 0

then so does A^T

Homework Equations

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.

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 $A^T$.

 

[Lord Anoobis]

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 $A^T$.

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