If A = 3&-2 4&-2 and I = 1&0 0&1 , find k so that A^2 = kA - 2I. - Vidyadip

Question

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

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Answer

Given: $\text{A}=\begin{bmatrix}3&-2\\4&-2\end{bmatrix} \text{and}\ \text{I}\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\text{A}^2=k\text{A}-2\text{I}\Rightarrow\begin{bmatrix}3&-2\\4&-2\end{bmatrix}\begin{bmatrix}3&-2\\4&-2\end{bmatrix}=k\begin{bmatrix}3&-2\\4&-2\end{bmatrix}-2\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}9-8&-6+4\\12-8&-8+4\end{bmatrix}=\begin{bmatrix}3k&-2k\\4k&-2k\end{bmatrix}-\begin{bmatrix}2&0\\0&2\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&-2\\4&-4\end{bmatrix}=\begin{bmatrix}3k-2&-2k-0\\4k-0&-2k-2\end{bmatrix}$
Equating corresponding entries, we have
$3k - 2 = 1 ⇒ 3k = 3$
$k = 1$
And 4k = 4 ⇒ k = 1 and -4 = -2k - 2
$\Rightarrow 2k = 2 \Rightarrow k = 1$
$\therefore$ k = 1

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