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Question:

If $A=\left[\begin{array}{rr}2 & -1 \\ 3 & 2\end{array}\right]$ and $B=\left[\begin{array}{rr}0 & 4 \\ -1 & 7\end{array}\right]$, find $3 A^{2}-2 B+1$

Solution:

Given : $A=\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]$

Now,

$A^{2}=A A$

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

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

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

$3 A^{2}-2 B+I$

$\Rightarrow 3 A^{2}-2 B+I=3\left[\begin{array}{cc}1 & -4 \\ 12 & 1\end{array}\right]-2\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]+\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow 3 A^{2}-2 B+I=\left[\begin{array}{cc}3 & -12 \\ 36 & 3\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]+\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow 3 A^{2}-2 B+I=\left[\begin{array}{cc}3-0+1 & -12-8+0 \\ 36+2+0 & 3-14+1\end{array}\right]$

$\Rightarrow 3 A^{2}-2 B+I=\left[\begin{array}{cc}4 & -20 \\ 38 & -10\end{array}\right]$

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