Question

Find the product of the matrices $A$ and $B$ where
$A=\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$

Hence, solve the following equations by matrix method
$x+y+2z=1$
$3x+2y+z=7$
$2x+y+3z=2$

Answer 1

$A=\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$

$\therefore AB=\left[\begin{array}{ccc}-5+3+6& -5+2+3& -10+1+9\\ 7+3-10& 7+2-5& 14+1-15\\ 1-3+2& 1-2+1& 2-1+3\end{array}\right]$

$AB=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$

$AB=4\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$AB=4I_3$

$\Rightarrow B\left(\frac{1}{4}A\right)=I_3$

$\Rightarrow B^{-1}=\frac{1}{4}A$

$\Rightarrow B^{-1}=\frac{1}{4}\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right]$

The given system of equation is
$x+y+2z=1$
$3x+2y+z=7$
$2x+y+3z=2$

This system can be written as
$\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 7\\ 2\end{array}\right]$ or $BX=C$

where $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],C=\left[\begin{array}{c}1\\ 7\\ 2\end{array}\right]$

As $B^{-1}$ exists, the given system has a unique solution
$X=B^{-1}C$

$\Rightarrow X=\frac{1}{4}\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right]\left[\begin{array}{c}1\\ 7\\ 2\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}-5+7+6\\ 7+7-10\\ 1-7+2\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ -1\end{array}\right]\Rightarrow x=2,y=1,z=-1$

Hence, the solution of given system of equation is $x=2,y=1,z=-1.$