# Multiplication of a Matrix by a Number (Sample Questions)

Example 1:
Express the following as single matrix.

$\text{(a)}3\left(\begin{array}{cc}1& -4\\ -3& 2\end{array}\right)+2\left(\begin{array}{cc}-3& 1\\ 3& -4\end{array}\right)$

$\text{(b)}2\left(\begin{array}{cc}-1& 0\\ 9& -4\end{array}\right)-4\left(\begin{array}{cc}-2& 2\\ 3& -6\end{array}\right)-\frac{1}{3}\left(\begin{array}{cc}-9& -3\\ -6& 15\end{array}\right)$

Solution:
$\begin{array}{l}\text{(a)}3\left(\begin{array}{cc}1& -4\\ -3& 2\end{array}\right)+2\left(\begin{array}{cc}-3& 1\\ 3& -4\end{array}\right)\\ =\left(\begin{array}{cc}3×1& 3×\left(-4\right)\\ 3×\left(-3\right)& 3×2\end{array}\right)+\left(\begin{array}{cc}2×\left(-3\right)& 2×1\\ 2×3& 2×\left(-4\right)\end{array}\right)\\ =\left(\begin{array}{cc}3& -12\\ -9& 6\end{array}\right)+\left(\begin{array}{cc}-6& 2\\ 6& -8\end{array}\right)\\ =\left(\begin{array}{cc}3+\left(-6\right)& -12+2\\ -9+6& 6-\left(-8\right)\end{array}\right)\\ =\left(\begin{array}{cc}-3& -10\\ -3& 14\end{array}\right)\end{array}$

$\begin{array}{l}\text{(b)}2\left(\begin{array}{cc}-1& 0\\ 9& -4\end{array}\right)-4\left(\begin{array}{cc}-2& 2\\ 3& -6\end{array}\right)-\frac{1}{3}\left(\begin{array}{cc}-9& -3\\ -6& 15\end{array}\right)\\ =\left(\begin{array}{cc}-2& 0\\ 18& -8\end{array}\right)-\left(\begin{array}{cc}-8& 8\\ 12& -24\end{array}\right)-\left(\begin{array}{cc}\frac{1}{3}×\left(-9\right)& \frac{1}{3}×\left(-3\right)\\ \frac{1}{3}×\left(-6\right)& \frac{1}{3}×15\end{array}\right)\\ =\left(\begin{array}{cc}-2& 0\\ 18& -8\end{array}\right)-\left(\begin{array}{cc}-8& 8\\ 12& -24\end{array}\right)-\left(\begin{array}{cc}-3& -1\\ -2& 5\end{array}\right)\\ =\left(\begin{array}{cc}-2-\left(-8\right)-\left(-3\right)& 0-8-\left(-1\right)\\ 18-12-\left(-2\right)& -8-\left(-24\right)-5\end{array}\right)\\ =\left(\begin{array}{cc}9& -7\\ 10& 11\end{array}\right)\end{array}$