Question

find the inverse of the matrix A =[1 - 1 1, 1 - 2 4, 1 2 2] by gauss Jordan method.

Answer 1

Answer on Question #73894 – Math – Quantitative Methods

Question

Find the inverse of the matrix $A = [1 - 1, 1 - 2, 4, 1, 2, 2]$ by gauss Jordan method.

Solution

$$A = \begin{array}{ccc} 1 & -1 & 1 \\ 1 & -2 & 4 \\ 1 & 2 & 2 \end{array} \qquad \qquad \begin{array}{ccc} 1 & 0 & 0 \\ l = 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}$$

1) $a_{1j}^{1} = \frac{a_{1j}}{a_{11}}$, $b_{1j}^{1} = \frac{b_{1s}}{a_{11}}$

$$A = \begin{array}{ccc} 1 & -1 & 1 \\ 1 & -2 & 4 \\ 1 & 2 & 2 \end{array} \qquad \qquad \begin{array}{ccc} 1 & 0 & 0 \\ l = 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}$$

2) $a_{2j}^{1} = a_{2j} - a_{1j}^{1}a_{21}, \ldots, a_{nj}^{1} = a_{nj} - a_{1j}^{1}a_{n1}$

$$b_{2j}^{1} = b_{2j} - b_{1j}^{1}a_{21}, \ldots, b_{nj}^{1} = b_{nj} - b_{1j}^{1}a_{n1}$$ $$A = \begin{array}{ccc} 1 & -1 & 1 \\ 0 & -1 & 3 \\ 0 & 3 & 1 \end{array} \qquad \qquad \begin{array}{ccc} 1 & 0 & 0 \\ l = -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}$$

3) $a_{2j}^{1} = \frac{a_{2j}}{a_{22}}$ $a_{ij}^{1} = a_{ij} - a_{2j}^{1}a_{i2}$

$$b_{2j}^{1} = \frac{b_{2s}}{a_{22}} \quad b_{ij}^{1} = b_{2j} - b_{ij}^{1}a_{i2}$$ $$A = \begin{array}{ccc} 1 & -1 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & 10 \end{array} \qquad \qquad \begin{array}{ccc} 1 & 0 & 0 \\ l = 1 & -1 & 0 \\ -4 & 3 & 1 \end{array}$$

4) $a_{3j}^{1} = \frac{a_{3j}}{a_{33}}$ $a_{ij}^{1} = a_{ij} - a_{3j}^{1}a_{i3}$

$$b_{3j}^{1} = \frac{b_{3s}}{a_{33}} \quad b_{ij}^{1} = b_{ij} - b_{3j}^{1}a_{i3}$$ $$A = \begin{array}{ccc} 1 & -1 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array} \qquad \qquad \begin{array}{ccc} l = 1 & 0 & 0 \\ 1 & -1 & 0 \\ -0.4 & 0.3 & 0.1 \end{array}$$

5) $a_{ij}^{1} = a_{ij} - a_{3j}a_{i3}$

$$b_{ij}^{1} = b_{ij} - b_{3j}a_{i3}$$ $$A = \begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \qquad \qquad \begin{array}{ccc} 1.4 & -0.3 & -0.1 \\ l = -0.2 & -0.1 & 0.3 \\ -0.4 & 0.3 & 0.1 \end{array}$$

6) $a_{ij}^{1} = a_{ij} - a_{2j}a_{i2}$

$$b_{ij}^{1} = b_{ij} - b_{2j}a_{i2}$$

I – inverse of the matrix [1 -1 1; 1 -2 4; 1 2 2]

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