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Question:

If $A=\left[\begin{array}{rr}1 & -3 \\ -4 & 3\end{array}\right]$, find $A^{2}-3 A-7 /$

Solution:

Given : $A=\left[\begin{array}{cc}1 & -3 \\ -4 & 3\end{array}\right]$

Now,

$A^{2}=A A$

$\Rightarrow A^{2}=\left[\begin{array}{cc}1 & -3 \\ -4 & 3\end{array}\right]\left[\begin{array}{cc}1 & -3 \\ -4 & 3\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{cc}1+12 & -3-6 \\ -4-12 & 12+9\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{cc}13 & -9 \\ -16 & 21\end{array}\right]$

$A^{2}-5 A-14 I$

$\Rightarrow A^{2}-3 A-7 I=\left[\begin{array}{cc}13 & -9 \\ -16 & 21\end{array}\right]-3\left[\begin{array}{cc}1 & -3 \\ -4 & 3\end{array}\right]-7\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow A^{2}-3 A-7 I=\left[\begin{array}{cc}13 & -9 \\ -16 & 21\end{array}\right]-\left[\begin{array}{cc}3 & -9 \\ -12 & 9\end{array}\right]-\left[\begin{array}{ll}7 & 0 \\ 0 & 7\end{array}\right]$

$\Rightarrow A^{2}-3 A-7 I=\left[\begin{array}{cc}3 & 0 \\ -4 & 5\end{array}\right]$

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