Find k, if A= [ ll 3 & -2 4 & -2 ] and A^2=K A-2

Question

Find $k$, if $A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$ and $A^2=K A-2$

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Answer

$\begin{aligned} & \mathrm{A}^2=\mathrm{kA}-2 \mathrm{l} \\ & \therefore \mathrm{AA}+2 \mathrm{l}=\mathrm{kA} \\ & \therefore \quad\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]+2\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\mathrm{k}\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right] \\ & \therefore \quad\left[\begin{array}{cc}9-8 & -6+4 \\ 12-8 & -8+4\end{array}\right]+\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]=\left[\begin{array}{ll}3 \mathrm{k} & -2 \mathrm{k} \\ 4 \mathrm{k} & -2 \mathrm{k}\end{array}\right] \\ & \therefore \quad\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]+\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]=\left[\begin{array}{ll}3 \mathrm{k} & -2 \mathrm{k} \\ 4 \mathrm{k} & -2 \mathrm{k}\end{array}\right] \\ & \therefore\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]=\left[\begin{array}{ll}3 k & -2 k \\ 4 k & -2 k\end{array}\right]\end{aligned}$

∴ By equality of matrices, we get 3k = 3 ∴ k = 1

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