Prove that for any 2 x 2 matrix A, A2 can be written in the linear form aA + bI
I = (identity matrix).
I, think I, know this is true, I just can't imagine how to prove it.
you should try to devlop this
\(\begin{pmatrix} a&b \\ c&d \end{pmatrix} ^{2} =e\begin{pmatrix} a&b \\ c&d \end{pmatrix} +f\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}\)
It should give the following equations:
\(\left\{\begin{matrix} a^2+bc=ae+f \\ ab+bd=eb \\ ac+dc=ec \\ bc+d^2=f+ed \end{matrix}\right.\)
By solving the second equation, you can easily find \(e=a+d\)
wich also work with the third equation.
with the first and the fourth we can find \(f=bc-ad\)
Therefor we have shown that this system as always a solution for e=a+d and f=bc-ad=-det(A)
