# Solve this

Question:

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

Solution:

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

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

$=\left[\begin{array}{cc}2 & 4 \\ 5 & -2 \\ 5 & 8\end{array}\right]$

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

$=\left[\begin{array}{cc}3 & 5 \\ -2 & 0 \\ 6 & -1\end{array}\right]+\left(\left[\begin{array}{cc}-1 & -1 \\ 7 & -2 \\ -1 & 9\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & 4 \\ 5 & -2 \\ 5 & 8\end{array}\right]$

Therefore, $(A+B)+C=A+(B+C)$

It is true for any matrix

Conclusion: $(\mathrm{A}+\mathrm{B})+\mathrm{C}=\mathrm{A}+(\mathrm{B}+\mathrm{C})$