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Matrices — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsMatrices4 Marks
Question
If $A=\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right]$ and $C=\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right]$ find $1) \ce{(AB)C , 2) A(BC)}$ Is $\ce{A(BC) = (AB)C} ?$
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Answer

$1)$
$\begin{array}{l}( AB ) C =\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right]\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right] \end{array} $
$ =\left[\begin{array}{ll}1(1)+3(4) & 1(2)+3(3) \\ 2(1)+4(4) & 2(2)+4(3)\end{array}\right]$
$=\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right]  $
$ =\left[\begin{array}{ll}13 & 11 \\ 18 & 16\end{array}\right]\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right]  $
$ =\left[\begin{array}{ll}13(4)+11(1) & 13(3)+11(2) \\ 18(4)+18(3) & 18(3)+16(2)\end{array}\right]  $
$ =\left[\begin{array}{ll}52+11 & 39+22 \\ 72+16 & 54+32\end{array}\right] \\ =\left[\begin{array}{ll}63 & 61 \\ 88 & 86\end{array}\right]$
$2)$
$\begin{array}{l} A(B C)=\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right]\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right] \end{array} $
$ =\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}1(4)+2(1) & 1(3)+2(2) \\ 4(4)+3(1) & 4(3)+3(2)\end{array}\right]  $
$ =\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}4+2 & 3+4 \\ 16+3 & 12+6\end{array}\right]  $
$ =\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}6 & 7 \\ 19 & 18\end{array}\right] $
$ =\left[\begin{array}{cc}6+57 & 7+54 \\ 12+76 & 14+72\end{array}\right]  $
$ =\left[\begin{array}{ll}63 & 61 \\ 88 & 86\end{array}\right]  $
$ \therefore A(B C)=( AB ) C $

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