Proving Matrix Equality: A^3+4A^2-2A+7i=0 Implies A^T Also Satisfies It?

[ahmed dawod]

1. show that if a square matrix A satisfies A^3+4A^2 -2A+7i=0 then so does A^T

i is the identity matrix

The Attempt at a Solution

I tried to multiply the equation by A^T but all in vain

 

[rasmhop]

Use:
$$(A+B)^T = A^T + B^T \qquad (AB)^T = A^TB^T \qquad (cA)^T = cA^T$$
to compute the transpose by using:
$$0 = 0^T = (A^3 + 4A^2 - 2A + 7I)^T$$

 

[ahmed dawod]

thanks
this is useful

problem solved