Question

Consider the matrix:
A =  1 3
 5 7
a.) compute A^2 and A^2 -8A
b.) find a degree 2 polynomial satisfied by A
c.) use the polynomial to find the inverse of A
d.) find a degree 2 polynomial satisfied by
B = (1/8)  -7 3
 5 1

Answer 1

I found part a to be:
\[A^2 =\left[\begin{matrix}16 & 24 \\ 40 & 64\end{matrix}\right]\]
\[A^2 +8A =\left[\begin{matrix}8 & 0 \\ 0 & 8\end{matrix}\right]\]
I need help on b, c, d

Answer 2

Do you know about the Cayley-Hamilton theorem? It seems like that has to do with what you're looking for.

Given a matrix \(M\) with characteristic polynomial \(p(\lambda)\), the C-H theorem says that \(p(M)={\bf0}\), where \(\bf0\) denotes the zero matrix.

The fact that \(A^2-8A=\begin{pmatrix}8&0\\0&8\end{pmatrix}=8I_2\) is useful, since this means
\[A^2-8A-8I_2=\begin{pmatrix}0&0\\0&0\end{pmatrix}={\bf0}\]as desired.

For part (c), you have
\[A^2-8A=A(A-8I_2)=8I_2\implies A-8I_2=8A^{-1}\implies A^{-1}=\frac{1}{8}A-I_2\]