### If A = [2-35-4-61],B=[-122203]and C=[43-14-21], Show that (A + B) + C = A + (B + C)

Exercise 2.2 | Q 1.2 | Page 46

#### QUESTION

If A = $\left[\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right],\text{B}=\left[\begin{array}{cc}-1& 2\\ 2& 2\\ 0& 3\end{array}\right]\text{and C}=\left[\begin{array}{cc}4& 3\\ -1& 4\\ -2& 1\end{array}\right]$, Show that (A + B) + C = A + (B + C)

#### SOLUTION

(A + B) + C = $\left\{\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right]+\left[\begin{array}{cc}-1& 2\\ 2& 2\\ 0& 3\end{array}\right]\right\}+\left[\begin{array}{cc}4& 3\\ -1& 4\\ -2& 1\end{array}\right]$

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

$\left[\begin{array}{cc}1& -1\\ 7& -2\\ -6& 4\end{array}\right]+\left[\begin{array}{cc}4& 3\\ -1& 4\\ -2& 1\end{array}\right]$

$\left[\begin{array}{cc}1+4& -1+3\\ 7-1& -2+4\\ -6-2& 4+1\end{array}\right]$

∴ (A+ B) + C = $\left[\begin{array}{cc}5& 2\\ 6& 2\\ -8& 5\end{array}\right]$      ....(i)

A + (B + C) = $\left[\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right]+\left\{\left[\begin{array}{cc}-1& 2\\ 2& 2\\ 0& \end{array}\right]+\left[\begin{array}{cc}4& 3\\ -1& 4\\ -2& 1\end{array}\right]\right\}$

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

$\left[\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right]\left[\begin{array}{cc}3& 5\\ 1& 6\\ -2& 4\end{array}\right]$

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

$\left[\begin{array}{cc}5& 2\\ 6& 2\\ -8& 5\end{array}\right]$         ....(ii)

From (i) and (ii), we get
(A + B) + C = A + (B + C).