Answer
a) The required matrix equation is:
$\left[ \begin{matrix}
1 & 1 & 2 \\
0 & 1 & 3 \\
3 & 0 & -2 \\
\end{matrix} \right]\left[ \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right]=\left[ \begin{matrix}
7 \\
-2 \\
0 \\
\end{matrix} \right]$
b) The solution set is: . $\left\{ -18,79,-27 \right\}$
Work Step by Step
(a)
Note that the coefficient matrix $ A $ contains all coefficients of the variables $ x,y\text{ and }z $.
The matrix $ X $ contains all the variables and the matrix $ B $ contains all the constant terms of the given linear system. Therefore, the matrix equation is given by:
$\left[ \begin{matrix}
1 & 1 & 2 \\
0 & 1 & 3 \\
3 & 0 & -2 \\
\end{matrix} \right]\left[ \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right]=\left[ \begin{matrix}
7 \\
-2 \\
0 \\
\end{matrix} \right]$
Hence, the required matrix equation is given by:
$\left[ \begin{matrix}
1 & 1 & 2 \\
0 & 1 & 3 \\
3 & 0 & -2 \\
\end{matrix} \right]\left[ \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right]=\left[ \begin{matrix}
7 \\
-2 \\
0 \\
\end{matrix} \right]$
(b)
Consider the system $ AX=B $
The solution of this system is $ X={{A}^{-1}}B $
Since it is given that the inverse of the matrix $ A=\left[ \begin{matrix}
1 & 1 & 2 \\
0 & 1 & 3 \\
3 & 0 & -2 \\
\end{matrix} \right]$ is $\left[ \begin{matrix}
-2 & 2 & 1 \\
9 & -8 & -3 \\
-3 & 3 & 1 \\
\end{matrix} \right]$
Therefore, $\begin{align}
& X={{A}^{-1}}B \\
& =\left[ \begin{matrix}
-2 & 2 & 1 \\
9 & -8 & -3 \\
-3 & 3 & 1 \\
\end{matrix} \right]\left[ \begin{matrix}
7 \\
-2 \\
0 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-2\left( 7 \right)+2\left( -2 \right)+1\left( 0 \right) \\
9\left( 7 \right)+\left( -8 \right)\left( -2 \right)+\left( -3 \right)\left( 0 \right) \\
\left( -3 \right)\left( 7 \right)+3\left( -2 \right)+1\left( 0 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-14-4 \\
63+16 \\
-21-6 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-18 \\
79 \\
-27 \\
\end{matrix} \right]
\end{align}$
This implies that $ x=-18,y=79\text{ and }z=-27$
Hence, the solution set is:
$\left\{ -18,79,-27 \right\}$
