Question
If $X=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|, Y=\left|\begin{array}{cc}-3 & 0 \\ 4 & 1\end{array}\right|$ and $Z=\left|\begin{array}{cc}5 & -1 \\ 3 & 2\end{array}\right|$, prove that $X(Y+Z)=X Y+X Z$
✓
Answer
$\begin{aligned} & X=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|_{2 \times 2} Y=\left|\begin{array}{cc}-3 & 0 \\ 4 & 1\end{array}\right|_{2 \times 2} Z=\left|\begin{array}{cc}5 & -1 \\ 3 & 2\end{array}\right|_{2 \times 2} \end{aligned}$
$ (Y+Z)=\left|\begin{array}{cc}-3 & 0 \\ 4 & 1\end{array}\right|+\left|\begin{array}{cc}5 & -1 \\ 3 & 2\end{array}\right|=\left|\begin{array}{cc}2 & -1 \\ 7 & 3\end{array}\right|_{2 \times 2} $
$ X(Y+Z)=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|\left|\begin{array}{cc}2 & -1 \\ 7 & 3\end{array}\right|$
$ \begin{aligned} & =\left|\begin{array}{cc} 2-14 & -1-6 \\ 2+21 & -1+9 \end{array}\right| \end{aligned} $
$ X(Y+Z)=\left|\begin{array}{cc} -12 & -7 \\ 23 & 8 \end{array}\right|_{2 \times 2} \ldots \ldots \ldots (1)$
$\begin{aligned} & X Y=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|\left|\begin{array}{cc}-3 & 0 \\ 4 & 1\end{array}\right| \end{aligned} $
$=\left|\begin{array}{cc}-3-8 & 0-2 \\ -3+12 & 0+3\end{array}\right| $
$ =\left|\begin{array}{cc}-11 & -2 \\ 9 & 3\end{array}\right|_{2 \times 2}$
$\begin{aligned} & X Z=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|\left|\begin{array}{cc}5 & -1 \\ 3 & 2\end{array}\right| \end{aligned} $
$ =\left|\begin{array}{cc}5-6 & -1-4 \\ 5+9 & -1+6\end{array}\right| $
$ =\left|\begin{array}{cc}-1 & -5 \\ 14 & 5\end{array}\right|_{2 \times 2}$
$ X Y+X Z=\left|\begin{array}{cc} -11 & -2 \\ 9 & 3 \end{array}\right|+\left|\begin{array}{cc} -1 & -5 \\ 14 & 5 \end{array}\right|$
$=\left|\begin{array}{cc} -12 & -7 \\ 23 & 8 \end{array}\right|_{2 \times 2} \ldots \ldots \text { (2) }$
from $(1)$ and $(2) X(Y+Z)=X Y+Y Z$
$ (Y+Z)=\left|\begin{array}{cc}-3 & 0 \\ 4 & 1\end{array}\right|+\left|\begin{array}{cc}5 & -1 \\ 3 & 2\end{array}\right|=\left|\begin{array}{cc}2 & -1 \\ 7 & 3\end{array}\right|_{2 \times 2} $
$ X(Y+Z)=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|\left|\begin{array}{cc}2 & -1 \\ 7 & 3\end{array}\right|$
$ \begin{aligned} & =\left|\begin{array}{cc} 2-14 & -1-6 \\ 2+21 & -1+9 \end{array}\right| \end{aligned} $
$ X(Y+Z)=\left|\begin{array}{cc} -12 & -7 \\ 23 & 8 \end{array}\right|_{2 \times 2} \ldots \ldots \ldots (1)$
$\begin{aligned} & X Y=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|\left|\begin{array}{cc}-3 & 0 \\ 4 & 1\end{array}\right| \end{aligned} $
$=\left|\begin{array}{cc}-3-8 & 0-2 \\ -3+12 & 0+3\end{array}\right| $
$ =\left|\begin{array}{cc}-11 & -2 \\ 9 & 3\end{array}\right|_{2 \times 2}$
$\begin{aligned} & X Z=\left|\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right|\left|\begin{array}{cc}5 & -1 \\ 3 & 2\end{array}\right| \end{aligned} $
$ =\left|\begin{array}{cc}5-6 & -1-4 \\ 5+9 & -1+6\end{array}\right| $
$ =\left|\begin{array}{cc}-1 & -5 \\ 14 & 5\end{array}\right|_{2 \times 2}$
$ X Y+X Z=\left|\begin{array}{cc} -11 & -2 \\ 9 & 3 \end{array}\right|+\left|\begin{array}{cc} -1 & -5 \\ 14 & 5 \end{array}\right|$
$=\left|\begin{array}{cc} -12 & -7 \\ 23 & 8 \end{array}\right|_{2 \times 2} \ldots \ldots \text { (2) }$
from $(1)$ and $(2) X(Y+Z)=X Y+Y Z$
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