Multiplication of Two Matrices (Examples)

Example 1:
Find the product of the following pairs of matrices.
$\begin{array}{l}\text{(a) }\left(152\right)\left(\begin{array}{l}2\\ 4\\ 3\end{array}\right)\\ \text{(b) }\left(\begin{array}{cc}2& 8\\ -3& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 4& -2\end{array}\right)\\ \text{(c) }\left(\begin{array}{l}-3\\ 5\end{array}\right)(2\text{ 6})\\ \text{(d) }\left(\begin{array}{cc}0& 4\\ -1& 3\end{array}\right)\left(\begin{array}{l}7\\ -2\end{array}\right)\\ \text{(e) }(7-4)\left(\begin{array}{cc}-2& 0\\ -1& 3\end{array}\right)\end{array}$

Solution:
(a)
$$\begin{array}{l}\left(152\right)\left(\begin{array}{l}2\\ 4\\ 3\end{array}\right)\leftarrow \boxed{\begin{array}{l}\text{Matrices analysis}\\ 1\times 3\text{ and 3}\times \text{1}\\ \searrow \swarrow \\ =1\times 1\text{ matrix}\end{array}}\\ =\left(1\times 2\oplus 5\times 4\oplus 2\times 3\right)\\ =\left(2+20+6\right)\\ =\left(28\right)\end{array}$$
(b)
$$\begin{array}{l}\left(\begin{array}{cc}2& 8\\ -3& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 4& -2\end{array}\right)\leftarrow \boxed{\begin{array}{l}\text{Matrices analysis}\\ 2\times 2\text{ and 2}\times 2\\ \searrow \swarrow \\ =2\times 2\text{ matrix}\end{array}}\\ =\left(\begin{array}{cc}2\times 1+8\times 4& 2\times 0+8\times -2\\ -3\times 1+1\times 4& -3\times 0+1\times -2\end{array}\right)\\ =\left(\begin{array}{cc}34& -16\\ 1& -2\end{array}\right)\end{array}$$
(c)
$$\begin{array}{l}\left(\begin{array}{l}-3\\ 5\end{array}\right)(2\text{ 6})\leftarrow \boxed{\begin{array}{l}\text{Matrices analysis}\\ 2\times 1\text{ and 1}\times 2\\ \searrow \swarrow \\ =2\times 2\text{ matrix}\end{array}}\\ =\left(\begin{array}{cc}-3\times 2& -3\times 6\\ 5\times 2& \text{ 5}\times 6\end{array}\right)\\ =\left(\begin{array}{cc}-6& -18\\ 10& 30\end{array}\right)\end{array}$$
(d)
$$\begin{array}{l}\left(\begin{array}{cc}0& 4\\ -1& 3\end{array}\right)\left(\begin{array}{l}7\\ -2\end{array}\right)\leftarrow \boxed{\begin{array}{l}\text{Matrices analysis}\\ 2\times 2\text{ and 2}\times 1\\ \searrow \swarrow \\ =2\times 1\text{ matrix}\end{array}}\\ =\left(\begin{array}{l}0\times 7+4\times -2\\ -1\times 7+3\times -2\end{array}\right)\\ =\left(\begin{array}{l}-8\\ -13\end{array}\right)\end{array}$$
(e)
$$\begin{array}{l}(7-4)\left(\begin{array}{cc}-2& 0\\ -1& 3\end{array}\right)\leftarrow \boxed{\begin{array}{l}\text{Matrices analysis}\\ 1\times 2\text{ and 2}\times 2\\ \searrow \swarrow \\ =1\times 2\text{ matrix}\end{array}}\\ =(7\times -2+\left(-4\times -1\right)\text{ 7}\times 0+\left(-4\times 3\right))\\ =(-14+4\text{ 0}-\text{12})\\ =(-10-\text{12})\end{array}$$

Example 2:

Find the values of m and n in each of the following matrix equations.

$\begin{array}{l}(a)\left(\begin{array}{l}3\\ m\end{array}\right)(1n)=\left(\begin{array}{cc}3& 12\\ -2& -8\end{array}\right)\\ (b)\left(\begin{array}{cc}m& 2\\ -3& 1\end{array}\right)\left(\begin{array}{l}2\\ n\end{array}\right)=\left(\begin{array}{l}12\\ 4+2n\end{array}\right)\\ (c)\left(\begin{array}{cc}m& -3\\ -1& 1\end{array}\right)\left(\begin{array}{cc}-1& 2\\ 4& n\end{array}\right)=\left(\begin{array}{cc}-14& -11\\ 5& 3\end{array}\right)\end{array}$

Solution:
$\begin{array}{l}(a)\left(\begin{array}{l}3\\ m\end{array}\right)(1\text{ 4})=\left(\begin{array}{cc}3& 12\\ -2& n\end{array}\right)\\ \left(\begin{array}{cc}3& 12\\ m& 4m\end{array}\right)=\left(\begin{array}{cc}3& 12\\ -2& n\end{array}\right)\\ m=-2,\\ 4m=n\\ 4(-2)=n\\ n=-8\\ \\ (b)\left(\begin{array}{cc}m& 2\\ -3& 1\end{array}\right)\left(\begin{array}{l}2\\ n\end{array}\right)=\left(\begin{array}{l}12\\ 4+2n\end{array}\right)\\ \left(\begin{array}{l}2m+2n\\ -6+n\end{array}\right)=\left(\begin{array}{l}12\\ 4+2n\end{array}\right)\\ -6+n=4+2n\\ n=-10\\ 2m+2n=12\\ 2m+2(-10)=12\\ 2m-20=12\\ 2m=32\\ m=16\\ \\ (c)\left(\begin{array}{cc}m& -3\\ -1& 1\end{array}\right)\left(\begin{array}{cc}-1& 2\\ 4& n\end{array}\right)=\left(\begin{array}{cc}-14& -11\\ 5& 3\end{array}\right)\\ \left(\begin{array}{cc}-m+(-12)& 2m+(-3n)\\ 1+4& -2+n\end{array}\right)=\left(\begin{array}{cc}-14& -11\\ 5& 3\end{array}\right)\\ -m-12=-14\\ -m=-2\\ m=2\\ -2+n=3\\ n=5\end{array}$