Question

Use the product of two matrices $A$ and $B$, where $A=\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$ to solve the following system of linear equations,

$x+y+2z=1$;

$3x+2y+z=7$;

$2x+y+3z=2$, for $x,y$ and $z$.

• A. $x=1,y=2,z=3$
• B. $x=2,y=1,z=-1$
• C. $x=-2,y=-1,z=-1$
• D. $x=0,y=1,z=2$

Answer 1

The correct option is B $x=2,y=1,z=-1$

$A=\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right]$
$B=\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$
$AB=\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$
$\Rightarrow AB=4I$, where $I$ is the identity matrix.
Given system of equations
$x+y+2z=1;3x+2y+z=7;2x+y+3z=2$

Converting these to matrix form,

$\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 7\\ 2\end{array}\right]$
Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],C=\left[\begin{array}{c}1\\ 7\\ 2\end{array}\right]$
$\Rightarrow BX=C$
$\Rightarrow \left(AB\right)X=AC$

$4IX=\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right]\left[\begin{array}{c}1\\ 7\\ 2\end{array}\right]$
$\Rightarrow X=\frac{1}{4}\left[\begin{array}{c}\begin{array}{c}8\\ 4\\ -4\end{array}\end{array}\right]$
$\Rightarrow \left[\begin{array}{c}\begin{array}{c}x\\ y\\ z\end{array}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}2\\ 1\\ -1\end{array}\end{array}\right]$

Hence, $x=2,y=1,z=-1$

Hence, option B.