Verify that A(B+C)=A B+A C, if A= [ cc 4 & -2 2 & 3 ], B= [ cc -1 & 1 3 & -2 ] and C= [ cc 4 & 1 2 & -1 ]

B + C & = [ rr -1 & 1 3 & -2 ]+ [ rr 4 & 1 2 & -1 ] & = [ rr -1+4 & 1+1 3+2 & -2-1 ]= [ rr 3 & 2 5 & -3 ] A ( B + C ) & = [ rr 4 & -2 2 & 3 ] [ rr 3 & 2 5 & -3 ] & = [ rr 12-10 & 8+6 6+15 & 4-9 ]= [ rr 2 & 14 21 & -5 ] Also, AB = [ rr 4 & -2 2 & 3 ] [ rr -1 & 1 3 & -2 ] = & [ rr -4-6 & 4+4 -2+9 & 2-6 ]= [ rr -10 & 8 7 & -4 ] AC & = [ rr 4 & -2 2 & 3 ] [ rr 4 & 1 2 & -1 ] & = [ rr 16-4 & 4+2 8+6 & 2-3 ]= [ rr 12 & 6 14 & -1 ] AB + AC & = [ rr -10 & 8 7 & -4 ]+ [ rr 12 & 6 14 & -1 ] & = [ rr -10+12 & 8+6 7+14 & -4-1 ]= [ rr 2 & 14 21 & -5 ] From (1) and (2), A(B+C)=A B+A C.

Verify that A(B+C)=A B+A C, if A= [ cc 4 & -2 2 & 3 ], B= [ cc -1…

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Question

Verify that $A(B+C)=A B+A C$, if $A=\left[\begin{array}{cc}4 & -2 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{cc}-1 & 1 \\ 3 & -2\end{array}\right]$ and $C=\left[\begin{array}{cc}4 & 1 \\ 2 & -1\end{array}\right]$

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Answer

$
\begin{aligned}
B + C & =\left[\begin{array}{rr}
-1 & 1 \\
3 & -2
\end{array}\right]+\left[\begin{array}{rr}
4 & 1 \\
2 & -1
\end{array}\right] \\
& =\left[\begin{array}{rr}
-1+4 & 1+1 \\
3+2 & -2-1
\end{array}\right]=\left[\begin{array}{rr}
3 & 2 \\
5 & -3
\end{array}\right] \\
\therefore A ( B + C ) & =\left[\begin{array}{rr}
4 & -2 \\
2 & 3
\end{array}\right]\left[\begin{array}{rr}
3 & 2 \\
5 & -3
\end{array}\right] \\
& =\left[\begin{array}{rr}
12-10 & 8+6 \\
6+15 & 4-9
\end{array}\right]=\left[\begin{array}{rr}
2 & 14 \\
21 & -5
\end{array}\right]
\end{aligned}
$
Also, $AB =\left[\begin{array}{rr}4 & -2 \\ 2 & 3\end{array}\right]\left[\begin{array}{rr}-1 & 1 \\ 3 & -2\end{array}\right]$
$
\begin{aligned}
= & \left[\begin{array}{rr}
-4-6 & 4+4 \\
-2+9 & 2-6
\end{array}\right]=\left[\begin{array}{rr}
-10 & 8 \\
7 & -4
\end{array}\right] \\
AC & =\left[\begin{array}{rr}
4 & -2 \\
2 & 3
\end{array}\right]\left[\begin{array}{rr}
4 & 1 \\
2 & -1
\end{array}\right] \\
& =\left[\begin{array}{rr}
16-4 & 4+2 \\
8+6 & 2-3
\end{array}\right]=\left[\begin{array}{rr}
12 & 6 \\
14 & -1
\end{array}\right] \\
\therefore AB + AC & =\left[\begin{array}{rr}
-10 & 8 \\
7 & -4
\end{array}\right]+\left[\begin{array}{rr}
12 & 6 \\
14 & -1
\end{array}\right] \\
& =\left[\begin{array}{rr}
-10+12 & 8+6 \\
7+14 & -4-1
\end{array}\right]=\left[\begin{array}{rr}
2 & 14 \\
21 & -5
\end{array}\right] \cdots
\end{aligned}
$
From (1) and $(2), A(B+C)=A B+A C$.