Answer
Sample:
(A+B)C=AC+BC
Work Step by Step
Sample:
Use A, B and C to verify (A+B)C=AC+BC
LHS:
$A+B=\left[\begin{array}{ll}
1+1 & 0+0\\
0+0 & 1+(-1)
\end{array}\right]=\left[\begin{array}{ll}
2 & 0\\
0 & 0
\end{array}\right]$
$(A+B)C=$
$=\left[\begin{array}{ll}
2 & 0\\
0 & 0
\end{array}\right]\left[\begin{array}{ll}
-1 & 0\\
0 & 1
\end{array}\right]=\left[\begin{array}{ll}
-2+0 & 0+0\\
0+0 & 0+0
\end{array}\right]=\left[\begin{array}{ll}
-2 & 0\\
0 & 0
\end{array}\right]$
RHS:
$AC=\left[\begin{array}{ll}
1 & 0\\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
-1 & 0\\
0 & 1
\end{array}\right]=\left[\begin{array}{ll}
-1 & 0\\
0 & 1
\end{array}\right]$
$BC=\left[\begin{array}{ll}
1 & 0\\
0 & -1
\end{array}\right]\left[\begin{array}{ll}
-1 & 0\\
0 & 1
\end{array}\right]=\left[\begin{array}{ll}
-1 & 0\\
0 & -1
\end{array}\right]$
$AC+BC=\left[\begin{array}{ll}
-1-1 & 0+0\\
0+0 & 1-1
\end{array}\right]=\left[\begin{array}{ll}
-2 & 0\\
0 & 0
\end{array}\right]$
Thus,
(A+B)C=AC+BC
