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Question:

If $A=\left[\begin{array}{rr}3 & -2 \\ 4 & -2\end{array}\right]$, find $k$ such that $A^{2}=k A-2 / 2$

Solution:

Given: $A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$

Now,

$A^{2}=A A$

$\Rightarrow A^{2}=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{ll}9-8 & -6+4 \\ 12-8 & -8+4\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{ll}1 & -2 \\ 4 & 4\end{array}\right]$

$A^{2}=k A-2 I_{2}$

$\Rightarrow\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=k\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]-2\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{ll}3 k & -2 k \\ 4 k & -2 k\end{array}\right]-\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{ll}3 k-2 & -2 k-0 \\ 4 k-0 & -2 k-2\end{array}\right]$

The corresponding elements of two equal matrices are equal.

$\therefore 1=3 k-2$

$\Rightarrow 1+2=3 k$

$\Rightarrow 3=3 k$

$\Rightarrow k=1$

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