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જે $\mathrm{A}^{\prime}=\left[\begin{array}{cc}3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right]$ અને $\mathrm{B}=\left[\begin{array}{ccc}-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right],$ હોય, તો $(\mathrm{A}+\mathrm{B})^{\prime}=\mathrm{A}^{\prime}+\mathrm{B}^{\prime}$.
Solution
It is known that $A=\left(A^{\prime}\right)^{\prime}$
Therefore, we have:
$A=\left[\begin{array}{lll}3 & -1 & 0 \\ 4 & 2 & 1\end{array}\right]$
$B^{\prime}=\left[\begin{array}{cc}-1 & 1 \\ 2 & 2 \\ 1 & 3\end{array}\right]$
$A+B=\left[\begin{array}{ccc}3 & -1 & 0 \\ 4 & 2 & 1\end{array}\right]+\left[\begin{array}{ccc}-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right]=\left[\begin{array}{lll}2 & 1 & 1 \\ 5 & 4 & 4\end{array}\right]$
$\therefore(A+B)^{\prime}=\left[\begin{array}{ll}2 & 5 \\ 1 & 4 \\ 1 & 4\end{array}\right]$
$A^{\prime}+B^{\prime}=\left[\begin{array}{cc}3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right]+\left[\begin{array}{cc}-1 & 1 \\ 2 & 2 \\ 1 & 3\end{array}\right]=\left[\begin{array}{ll}2 & 5 \\ 1 & 4 \\ 1 & 4\end{array}\right]$
Thus, we verified that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$
