Question

a) Find the inverse of the matrix











−
−
=
1 2 2
1 2 4
1 1 1
A
using Gauss-Jordon method.

Answer 1

Answer on Question #46679 – Math – Linear Algebra

a) Find the inverse of the matrix

1 2 2

1 2 4

1 1 1

A

using Gauss-Jordon method.

Solution.

To calculate inverse matrix, we need to do the following steps.

- Set the matrix and append the identity matrix of the same dimension to it.

- Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one).

As a result, we will get the inverse calculated on the right.

If a determinant of the main matrix is zero, inverse doesn't exist.

Thus:

$$\left( \begin{array}{ccc} 1 & 2 & 2 \\ 1 & 2 & 4 \\ 1 & 1 & 1 \end{array} \right) \to \left( \begin{array}{cccc} 1 & 2 & 2 & 1 & 0 & 0 \\ 1 & 2 & 4 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \right) \to \left( \begin{array}{cccc} 1 & 2 & 2 & 1 & 0 & 0 \\ 1 & 2 & 4 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \right) \to \text{(Subtract the}$$ $$1 \text{st row from the 2nd and 3rd)} \to \left( \begin{array}{cccc} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 0 & 2 & -1 & 1 & 0 \\ 0 & -1 & -1 & -1 & 0 & 1 \end{array} \right) \to$$

(Find the pivot in the 2nd column (inversing the sign in the whole row) and swap

the 3rd and the 2nd rows) $\to$ $\begin{pmatrix} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 0 & 2 & -1 & 1 & 0 \end{pmatrix} \to \text{(Multiply the 2nd row by 2 and subtract the 2nd row from the 1st row)}$\to$

$$\rightarrow \left(\begin{array}{cccc}1 & 0 & 0 & -1 & 0 & -2\\0 & 2 & 2 & 2 & 0 & 2\\0 & 0 & 2 & -1 & 1 & 0\end{array}\right)\rightarrow (\text{Subtract the 3rd row from the 2nd row and divide } 2^{\text{nd}}\text{ and } 3^{\text{rd}}\text{ rows by 2})$$ $$\rightarrow \left(\begin{array}{cccc}1 & 0 & 0 & -1 \\ 0 & 1 & 0 & \frac{3}{2} \\ 0 & 0 & 1 & -\frac{1}{2} \end{array}\right)$$ $$\text{So, } \left(\begin{array}{ccc}1 & 2 & 2 \\ 1 & 2 & 4 \\ 1 & 1 & 1\end{array}\right)^{-1} = \left(\begin{array}{ccc}-\frac{1}{2} & \frac{0}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} & 1 \\ -\frac{1}{2} & \frac{1}{2} & 0\end{array}\right)$$

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