Question
If $A=\left[\begin{array}{ccc}4 & 3 & 1 \\ 2 & 3 & -8 \\ 1 & 0 & -4\end{array}\right], B=\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 9 & 2 \\ -7 & 1 & -1\end{array}\right]$ and $C=\left[\begin{array}{ccc}8 & 3 & 4 \\ 1 & -2 & 3 \\ 2 & 4 & -1\end{array}\right]$ then verify that $A+(B+C)=(A+B)+C$

Answer

$\begin{aligned} & B+C=\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 9 & 2 \\ -7 & 1 & -1\end{array}\right]+\left[\begin{array}{ccc}8 & 3 & 4 \\ 1 & -2 & 3 \\ 2 & 4 & -1\end{array}\right] \\ & =\left[\begin{array}{ccc}10 & 6 & 8 \\ 2 & 7 & 5 \\ -5 & 5 & -2\end{array}\right] \\ & A+(B+C)=\left[\begin{array}{ccc}4 & 3 & 1 \\ 2 & 3 & -8 \\ 1 & 0 & -4\end{array}\right]+\left[\begin{array}{ccc}10 & 6 & 8 \\ 2 & 7 & 5 \\ -5 & 5 & -2\end{array}\right] \\ & =\left[\begin{array}{ccc}14 & 9 & 9 \\ 4 & 10 & -3 \\ -4 & 5 & -6\end{array}\right] \\ & (A+B)=\left[\begin{array}{lll}4 & 3 & 1 \\ 2 & 3 & 8 \\ 1 & 0 & 4\end{array}\right]+\left[\begin{array}{ccc}2 & 3 & 4 \\ 1 & 9 & 2 \\ -7 & 1 & -1\end{array}\right] \\ & =\left[\begin{array}{ccc}6 & 6 & 5 \\ 3 & 12 & -6 \\ -6 & 1 & -5\end{array}\right] \\ & \end{aligned}$
$
\begin{aligned}
& (A+B)+C=\left[\begin{array}{ccc}
6 & 6 & 5 \\
3 & 12 & -6 \\
-6 & 1 & -5
\end{array}\right]+\left[\begin{array}{ccc}
8 & 3 & 4 \\
1 & -2 & 3 \\
2 & 4 & -1
\end{array}\right] \\
& =\left[\begin{array}{ccc}
14 & 9 & 9 \\
4 & 10 & -3 \\
-4 & 5 & -6
\end{array}\right] \ldots .(2)
\end{aligned}
$

From (1) and (2) we get
$
A+(B+C)=(A+B)+C
$

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