If A = [ cc 1 & 2 -2 & 3 ], B = [ ll 2 & 1 2 & 3 ] C = [ cc -3 & … - Vidyadip

Question

If $A =\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right], B =\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right] C =\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right]$ verify that
$A(B + C) = AB + AC$.

✓

Answer

$B+C=\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right]+\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right] $
$ =\left[\begin{array}{ll}2-3 & 1+1 \\ 2+2 & 3+0\end{array}\right]=\left[\begin{array}{cc}-1 & 2 \\ 4 & 3\end{array}\right]$
$A(B+C)=\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]\left[\begin{array}{cc}-1 & 2 \\ 4 & 3\end{array}\right] $
$ =\left[\begin{array}{cc}-1+8 & 2+6 \\ 2+12 & -4+9\end{array}\right]=\left[\begin{array}{cc}7 & 8 \\ 14 & 5\end{array}\right]$
Now $ A B=\left[\begin{array}{ll}6 & 7 \\ 2 & 7\end{array}\right] \\ A C=\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right] $
$ =\left[\begin{array}{cc}-3+4 & 1+0 \\ 6+6 & -2+0\end{array}\right]=\left[\begin{array}{cc}1 & 1 \\ 12 & -2\end{array}\right]$
$ AB + AC =\left[\begin{array}{ll}6 & 7 \\ 2 & 7\end{array}\right]+\left[\begin{array}{cc}1 & 1 \\ 12 & -2\end{array}\right] $
$ =\left[\begin{array}{cc}6+1 & 7+1 \\ 2+12 & 7-2\end{array}\right] $
$ AB + AC =\left[\begin{array}{cc}7 & 8 \\ 14 & 5\end{array}\right] $
Hence $A ( B + C )= AB + AC .$

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