Question

If A = $\left[\begin{array}{cc}9& 1\\ 7& 8\end{array}\right]$, B = $\left[\begin{array}{cc}1& 5\\ 7& 12\end{array}\right]$, find matrix C such that 5A + 3B + 2C is a null matrix.

Answer 1

$$\begin{gathered} \mathrm{Given}:5A+3B+2C=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow 5\left[\begin{array}{cc}9& 1\\ 7& 8\end{array}\right]+3\left[\begin{array}{cc}1& 5\\ 7& 12\end{array}\right]+2C=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}45& 5\\ 35& 40\end{array}\right]+\left[\begin{array}{cc}3& 15\\ 21& 36\end{array}\right]+2C=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}45+3& 5+15\\ 35+21& 40+36\end{array}\right]+2C=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}48& 20\\ 56& 76\end{array}\right]+2C=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow 2C=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]-\left[\begin{array}{cc}48& 20\\ 56& 76\end{array}\right] \\ \Rightarrow C=\frac{1}{2}\left[\begin{array}{cc}-48& -20\\ -56& -76\end{array}\right] \\ \Rightarrow C=\left[\begin{array}{cc}-24& -10\\ -28& -38\end{array}\right] \end{gathered}$$