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