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Matrices — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsMatrices3 Marks
Question
Given martices $A=\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}3 & 4 \\ -1 & -2\end{array}\right], C=\left[\begin{array}{cc}-3 & 1 \\ 0 & -2\end{array}\right]$ Find the products of (i) $A B C$ (ii) $A C B$ and state whether they are equal.
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Answer

$\begin{aligned} & A=\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right] \\ & B=\left[\begin{array}{cc}3 & 4 \\ -1 & -2\end{array}\right] \\ & C=\left[\begin{array}{cc}-3 & 1 \\ 0 & -2\end{array}\right] \\ & A B C=\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right] \times\left[\begin{array}{cc}3 & 4 \\ -1 & -2\end{array}\right] \times\left[\begin{array}{cc}-3 & 1 \\ 0 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}6-1 & 8-2 \\ 12-2 & 16-4\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 0 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}5 & 6 \\ 10 & 12\end{array}\right] \times\left[\begin{array}{cc}-3 & 1 \\ 0 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}-15+0 & 5-12 \\ -30+0 & 10-24\end{array}\right] \\ & =\left[\begin{array}{cc}-15 & -7 \\ -30 & -14\end{array}\right] \\ & A C B=\left[\begin{array}{cc}2 & 1 \\ 4 & 2\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 0 & -2\end{array}\right] \times\left[\begin{array}{cc}3 & 4 \\ -1 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}-6+0 & 2-2 \\ -12+10 & 4-4\end{array}\right] \times\left[\begin{array}{cc}3 \\ -1 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}-6 & 0 \\ -12 & 0\end{array}\right] \times\left[\begin{array}{cc}3 & 4 \\ -1 & -2\end{array}\right] \\ & =\left[\begin{array}{cc}-18+0 & -24+0 \\ -36+0 & -48+0\end{array}\right] \\ & =\left[\begin{array}{ll}-18 & -24 \\ -36 & -48\end{array}\right] \\ & \therefore A B C \neq A C B\end{aligned}$

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