Question

If A=$\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ and I=$\begin{bmatrix}1&0\\0&1\end{bmatrix}$,find k so that A²=kA-2I

Answer 1

A²=A.A

$\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$$\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ 

=$\begin{bmatrix}3(3)+(-2)(4)&3(-2)+(-2)(-2)\\4(3)+(-2)(4)&4(-2)+(-2)(-2)\end{bmatrix}$=$\begin{bmatrix}1&-2\\4&-4\end{bmatrix}$

Now A²=kA-2I
$\Rightarrow \begin{bmatrix}1&-2\\4&-4\end{bmatrix}$=$\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$-2$\begin{bmatrix}1&0\\0&1\end{bmatrix}$

$\Rightarrow$ $\begin{bmatrix}1&-2\\4&-4\end{bmatrix}$=$\begin{bmatrix}3k&-2k\\4k&-2k\end{bmatrix}$-2$\begin{bmatrix}1&0\\0&1\end{bmatrix}$

$\Rightarrow$ $\begin{bmatrix}1&-2\\4&-4\end{bmatrix}$=$\begin{bmatrix}3k-2&-2k\\4k-2k&-2\end{bmatrix}$
Comparing the corresponding elements, we have:
3k-2=1
3k=2
$\Rightarrow$ k=1

Thus, the value of k is 1.