Question

If $A=\left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right],B=\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right],C=\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]$, verify the following :

(ii) $\left(A+B\right)+C=A+\left(B+C\right)$

Answer 1

Step1: Calculation of the matrix $\left(A+B\right)+C$.

The addition of two or more matrices of same order can be done by adding the corresponding elements of matrices.

$\begin{array}{rcl}\left(A+B\right)+C& =& \left(\left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]+\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]\right)+\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]\\ & =& \left[\begin{array}{cc}2-3& 1+2\\ -1+4& 3+1\end{array}\right]+\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]\\ & =& \left[\begin{array}{cc}-1& 3\\ 3& 4\end{array}\right]+\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]\\ & =& \left[\begin{array}{cc}-1-3& 3+2\\ 3+4& 4+1\end{array}\right]\\ & =& \left[\begin{array}{cc}-4& 5\\ 7& 5\end{array}\right]\end{array}$

Step2: Calculation of the matrix $A+\left(B+C\right)$.

$\begin{array}{rcl}A+\left(B+C\right)& =& \left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]+\left(\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]+\left[\begin{array}{cc}-3& 2\\ 4& 1\end{array}\right]\right)\\ & =& \left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]+\left[\begin{array}{cc}-3-3& 2+2\\ 4+4& 1+1\end{array}\right]\\ & =& \left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]+\left[\begin{array}{cc}-6& 4\\ 8& 2\end{array}\right]\\ & =& \left[\begin{array}{cc}2-6& 1+4\\ -1+8& 3+2\end{array}\right]\\ & =& \left[\begin{array}{cc}-4& 5\\ 7& 5\end{array}\right]\end{array}$

Conclusion: The relation $\left(A+B\right)+C=A+\left(B+C\right)$ is verified.