How do I solve this matrix agebra question? A^2 - 5A - 6I

Question

A and B are matrices; I is an identity matrix

Also, what are some general rules I can follow when factorizing matrices?

Thank you :)

Answer 1

I assume you want to solve

 

A^2 - 5A - 6I = 0 (the matrix 0)

 

If you try factoring this like it was just numbers, you get

 

(A + 1I)(A - 6I) = (A + I)(A - 6I) = A^2 - 6AI + IA - 6I^2

 

Recall AI = IA = A and I^2 = I

 

Then

 

(A + 1I)(A - 6I) = A^2 - 6A + A - 6I = A^2 - 5A - 6I

 

So this appears to be a valid factorization.

 

Then A = -I and A = 6I are solutions.