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Matrices — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsMatrices3 Marks
Question
If $A=\left[\begin{array}{cc}1 & -2 \\ 2 & -1\end{array}\right]$ and $B=\left[\begin{array}{cc}3 & 2 \\ -2 & 1\end{array}\right]$ Find $2 B-A^2$
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Answer

$\begin{aligned} & A=\left[\begin{array}{cr}1 & -2 \\ 2 & -1\end{array}\right] \\ & B=\left[\begin{array}{cr}3 & 2 \\ -2 & 1\end{array}\right] \\ & 2 B=2\left[\begin{array}{cr}3 & 2 \\ -2 & 1\end{array}\right] \\ & =\left[\begin{array}{cr}6 & 4 \\ -4 & 2\end{array}\right] \\ & A^2= A \times A =\left[\begin{array}{ll}1 & -2 \\ 2 & -1\end{array}\right]\left[\begin{array}{cc}1 & -2 \\ 2 & -1\end{array}\right] \\ & =\left[\begin{array}{cc}1-4 & -2+2 \\ 2-2 & -4+1\end{array}\right] \\ & =\left[\begin{array}{cc}-3 & 0 \\ 0 & -3\end{array}\right] \\ & \therefore 2 B - A ^2=\left[\begin{array}{cc}6 & 4 \\ -4 & 2\end{array}\right]-\left[\begin{array}{cc}3- & 0 \\ 0 & -3\end{array}\right] \\ & =\left[\begin{array}{cc}6-(-3) & 4-0 \\ -4-0 & 2-(-3)\end{array}\right] \\ & =\left[\begin{array}{cc}6+3 & 4 \\ -4 & 2+3\end{array}\right] \\ & =\left[\begin{array}{cc}9 & 4 \\ -4 & 5\end{array}\right] .\end{aligned}$

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