‹ Back to the paper
Solve the following.
Evaluate (3 -2 3; 2 1 -1; 4 -3 2) (-1 -5 -1; -8 -6 9; -10 1 7). Hence, solve the system of…
Evaluate $\begin{pmatrix} 3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2 \end{pmatrix} \begin{pmatrix} -1 & -5 & -1 \\ -8 & -6 & 9 \\ -10 & 1 & 7 \end{pmatrix}$.
Hence, solve the system of equations:
$3x - 2y + 3z = 8$
$2x + y - z = 1$
$4x - 3y + 2z = 4$
Answer
Answer
Matrix: $AB = \begin{bmatrix} -17 & 0 & 0 \\ 0 & -17 & 0 \\ 0 & 0 & -17 \end{bmatrix}$
Solution: $x = 1$, $y = 2$, $z = 3$
$AB = \begin{pmatrix} 3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2 \end{pmatrix}\begin{pmatrix} -1 & -5 & -1 \\ -8 & -6 & 9 \\ -10 & 1 & 7 \end{pmatrix} = \begin{pmatrix} -17 & 0 & 0 \\ 0 & -17 & 0 \\ 0 & 0 & -17 \end{pmatrix} = -17I$.
So $A^{-1} = -\frac{1}{17}B$.
The system is $AX = C$ with $C = (8, 1, 4)^T$, so $X = A^{-1}C = -\frac{1}{17}BC$.
$BC = \begin{pmatrix} -8-5-4 \\ -64-6+36 \\ -80+1+28 \end{pmatrix} = \begin{pmatrix} -17 \\ -34 \\ -51 \end{pmatrix}$
$X = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.
Hence $x = 1$, $y = 2$, $z = 3$.
Final answer: $x = 1,\ y = 2,\ z = 3$