Question

The matrix $A^{2} + 4A - 5I$, where I is identity matrix and $A = \begin{bmatrix}1&2\\ 4&-3\end{bmatrix} $, equals

• A. $4\begin{bmatrix}2&1\\ 2&0\end{bmatrix}$
• B. $4\begin{bmatrix}0&-1\\ 2&2\end{bmatrix}$
• C. $32\begin{bmatrix}2&1\\ 2&0\end{bmatrix}$
• D. $32\begin{bmatrix}1&1\\ 1&0\end{bmatrix}$

Answer 1

Answer: (A)

$A^{2} + 4A - 5I = A \times A + 4A - 5I $
$ = \begin{bmatrix}1&2\\ 4&-3\end{bmatrix} \times\begin{bmatrix}1&2\\ 4&-3\end{bmatrix} + 4 \begin{bmatrix}1&2\\ 4&-3\end{bmatrix} -5 \begin{bmatrix}1&0\\ 0&1\end{bmatrix} $
$ = \begin{bmatrix}9&-4\\ -8&17\end{bmatrix}+\begin{bmatrix}4&8\\ 16&-12\end{bmatrix} - \begin{bmatrix}5&0\\ 0&5\end{bmatrix}$
$ = \begin{bmatrix}9+4-5&-4+8-0\\ -8+16-0&17-12-5\end{bmatrix} = \begin{bmatrix}8&4\\ 8&0\end{bmatrix} $
$ = 4 \begin{bmatrix}2&1\\ 2&0\end{bmatrix}$