Use inverse matrix to solve the system of equations
\[
\begin{array}{c}
7 x+2 y+z=21 \\
3 y-z=5 \\
-3 x+4 y-2 z=-1
\end{array}
\]
Jumerical Methods
Page
UMaT, Department of Mathematics
Solution:
In matrix form these equations can be written as
\[
\left(\begin{array}{ccc}
7 & 2 & 1 \\
0 & 3 & -1 \\
-3 & 4 & -2
\end{array}\right)\left(\begin{array}{l}
x \\
y \\
x
\end{array}\right)=\left(\begin{array}{c}
21 \\
5 \\
-1
\end{array}\right)
\]
Recall that when \( A x=b \), then \( x=A^{-1} b \) so
\[
\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)=\left(\begin{array}{ccc}
-2 & 8 & -5 \\
3 & -11 & 7 \\
9 & -34 & 21
\end{array}\right)\left(\begin{array}{c}
21 \\
5 \\
-1
\end{array}\right)=\left(\begin{array}{c}
-42+40+5 \\
63-55-7 \\
189-170-21
\end{array}\right)=\left(\begin{array}{c}
3 \\
1 \\
-2
\end{array}\right)
\]
Hence \( x=3, y=1, z=-2 \).