Answer
- The inverse of the given matrix is:
\[ \left( \begin{array}{ccc}
\frac 23 & \frac 13 & -\frac {10}3 & \frac {10}3\\
\frac 13 & -\frac 13 & \frac 43 & -\frac 73 \\
-\frac 23 & \frac 23 & -\frac 23 & \frac 53 \\
\frac 1 3 & -\frac 13 & \frac 13 & -\frac 13 \end{array} \right)\]
Work Step by Step
1. Put an identity $I$ matrix on the right of the given matrix, to get a $4 \times 8$ one.
\[ \left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 1 & 0 & 0 & 0\\
1 & 3 & 4 & 2 & 0 & 1 & 0 & 0\\
0 & 1 & 2 & 3 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 2 & 0 & 0 & 0 & 1 \end{array} \right)\]
2. Row-reduce the whole matrix:
$R_2 = R_2 - R_1$:
\[ \left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 1 & 0 & 0 & 0\\
0 & 1 & 1 & -2 & -1 & 1 & 0 & 0\\
0 & 1 & 2 & 3 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 2 & 0 & 0 & 0 & 1 \end{array} \right)\]
$R_1 = R_1 - 2R_2$
$R_3 = R_3 - R_2$
\[ \left( \begin{array}{ccc}
1 & 0 & 1 & 8 & 3 & -2 & 0 & 0\\
0 & 1 & 1 & -2 & -1 & 1 & 0 & 0\\
0 & 0 & 1 & 5 & 1 & -1 & 1 & 0 \\
0 & 0 & 1 & 2 & 0 & 0 & 0 & 1 \end{array} \right)\]
$R_1 = R_1 - R_3$
$R_2 = R_2 - R_3$
$R_4 = R_4 - R_3$
\[ \left( \begin{array}{ccc}
1 & 0 & 0 & 10 & 4 & -3 & 0 & 0\\
0 & 1 & 0 & -7 & -2 & 2 & -1 & 0\\
0 & 0 & 1 & 5 & 1 & -1 & 1 & 0 \\
0 & 0 & 0 & -3 & -1 & 1 & -1 & 1 \end{array} \right)\]
$R_1 = R_1 + \frac{10} 3R_4$
$R_2 = R_2 - \frac 73R_4$
$R_3 = R_3 + \frac 53 R_4$
\[ \left( \begin{array}{ccc}
1 & 0 & 0 & 0 & \frac 23 & \frac 13 & -\frac {10}3 & \frac {10}3\\
0 & 1 & 0 & 0 & \frac 13 & -\frac 13 & \frac 43 & -\frac 73 \\
0 & 0 & 1 & 0 & -\frac 23 & \frac 23 & -\frac 23 & \frac 53 \\
0 & 0 & 0 & -3 & -1 & 1 & -1 & 1 \end{array} \right)\]
$R_4 = -\frac 13R_4$
\[ \left( \begin{array}{ccc}
1 & 0 & 0 & 0 & \frac 23 & \frac 13 & -\frac {10}3 & \frac {10}3\\
0 & 1 & 0 & 0 & \frac 13 & -\frac 13 & \frac 43 & -\frac 73 \\
0 & 0 & 1 & 0 & -\frac 23 & \frac 23 & -\frac 23 & \frac 53 \\
0 & 0 & 0 & 1 & \frac 1 3 & -\frac 13 & \frac 13 & -\frac 13 \end{array} \right)\]
- Thus, the inverse of the given matrix is:
\[ \left( \begin{array}{ccc}
\frac 23 & \frac 13 & -\frac {10}3 & \frac {10}3\\
\frac 13 & -\frac 13 & \frac 43 & -\frac 73 \\
-\frac 23 & \frac 23 & -\frac 23 & \frac 53 \\
\frac 1 3 & -\frac 13 & \frac 13 & -\frac 13 \end{array} \right)\]