How do I do
If A = $\displaystyle \begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}$, write A^2 in the form pA+qI where p and q are scalars?
How do I convert a square matrix to _A+_I?
Really need help, thank you so so much!
I see no reason not to just do this by straightforward calculation.
Since the problem asks about $\displaystyle A^2$, the obvious first thing to do is to square A!
$\displaystyle \begin{bmatrix}3 & 2 \\ -2 & -1\end{bmatrix}\begin{bmatrix}3 & 2 \\ -2 & -1\end{bmatrix}= \begin{bmatrix}5 & 4 \\ -4 & -3 \end{bmatrix}$
I presume you know that I is the identity matrix: $\displaystyle \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ so that, for any q, $\displaystyle qI= \begin{bmatrix}q & 0 \\ 0 & q\end{bmatrix}$ and then $\displaystyle A^2- qI= \begin{bmatrix}5- q & 4 \\ -4 & -3- q\end{bmatrix}$ and that must be equal to $\displaystyle pA= \begin{bmatrix}3p & 2p \\ -2p & -p\end{bmatrix}$.
That is, we have the four equations, 5- q= 3p, 4= 2p, -4= -2p, and -3- q= -p. Solve those equations for p and q.