Question

If $A=(\begin{array}{ccc}4& 1& 2\\ 1& -2& 3\\ 0& 3& 2\end{array}),B=(\begin{array}{ccc}2& 0& 4\\ 6& 2& 8\\ 2& 4& 6\end{array})$ and $C=(\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array})$, then
verify that $A+(B+C)=(A+B)+C$.

Answer 1

$A=\left[\begin{array}{ccc}4& 1& 2\\ 1& -2& 3\\ 0& 3& 2\end{array}\right],B=\left[\begin{array}{ccc}2& 0& 4\\ 6& 2& 8\\ 2& 4& 6\end{array}\right],C=\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right]$

$\Rightarrow A+(B+C)=(A+B)+C$

$\Rightarrow L.H.S=A+(B+C)$

$=\left[\begin{array}{ccc}4& 1& 2\\ 1& -2& 3\\ 0& 3& 2\end{array}\right]+[\left[\begin{array}{ccc}2& 0& 4\\ 6& 2& 8\\ 2& 4& 6\end{array}\right]+\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right]]$

$=\left[\begin{array}{ccc}4& 1& 2\\ 1& -2& 3\\ 0& 3& 2\end{array}\right]+\left[\begin{array}{ccc}3& 2& 1\\ 11& 2& 10\\ 3& 3& 7\end{array}\right]$

$=\left[\begin{array}{ccc}7& 3& 3\\ 12& 0& 13\\ 3& 6& 9\end{array}\right]$

$\Rightarrow R.H.S=(A+B)+C$

$=[\left[\begin{array}{ccc}4& 1& 2\\ 1& -2& 3\\ 0& 3& 2\end{array}\right]+\left[\begin{array}{ccc}2& 0& 4\\ 6& 2& 8\\ 2& 4& 6\end{array}\right]]+\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right]$

$=\left[\begin{array}{ccc}6& 1& 6\\ 7& 0& 11\\ 2& 7& 8\end{array}\right]+\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right]$

$=\left[\begin{array}{ccc}7& 3& 3\\ 12& 0& 13\\ 3& 6& 9\end{array}\right]$

$\Rightarrow L.H.S=R.H.S$

Hence, verified.