Answer
- The inverse of the given matrix is:
\[ \left( \begin{array}{ccc}
1 & -\frac12 & -\frac 52\\
0 & \frac 14 & -\frac 14\\
0 & 0 & 1\end{array} \right)\]
Work Step by Step
1. Put an identity $I$ matrix on the right of the given matrix, to get a $2 \times 4$ one.
\[ \left( \begin{array}{ccc}
1 & 2 & 3 & 1 & 0 & 0\\
0 & 4 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 0 & 1\end{array} \right)\]
2. Row-reduce the whole matrix:
$R_1 = 2R_1 - R_2$:
\[ \left( \begin{array}{ccc}
2 & 0 & 5 & 2 & -1 & 0\\
0 & 4 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 0 & 1\end{array} \right)\]
$R_1 = R_1 - 5R_3$
$R_2 = R_2 - R_3$
\[ \left( \begin{array}{ccc}
2 & 0 & 0 & 2 & -1 & -5\\
0 & 4 & 0 & 0 & 1 & -1\\
0 & 0 & 1 & 0 & 0 & 1\end{array} \right)\]
$R_1 = \frac 1 2 R_1$
$R_2 = \frac 1 4 R_2$
\[ \left( \begin{array}{ccc}
1 & 0 & 0 & 1 & -\frac12 & -\frac 52\\
0 & 1 & 0 & 0 & \frac 14 & -\frac 14\\
0 & 0 & 1 & 0 & 0 & 1\end{array} \right)\]
- Thus, the inverse of the given matrix is:
\[ \left( \begin{array}{ccc}
1 & -\frac12 & -\frac 52\\
0 & \frac 14 & -\frac 14\\
0 & 0 & 1\end{array} \right)\]
