TSSFL TECHNOLOGY STACK

Posted: **Sat Jan 02, 2016 4:10 pm**

Example: Apply elementary row operations to find the inverse of the matrix \[A = \begin{bmatrix} -1 & 3 & 2\\1&-2& -3\\2&-4&-2 \end{bmatrix}\]

Solution

Apply successively elementary row operations

$r_1 + r_2 \rightarrow r_2, \ 2r_1 + r_3 \rightarrow r_3, \ -2r_2 + r_3 \rightarrow r_3, \ \frac{1}{2}r_3+ r_1 \rightarrow r_1, \frac{1}{4}r_3 + r_2 \rightarrow r_2, \ 3r_2 + r_1 \rightarrow r_1.$

Thus

$[A:I] = \begin{bmatrix}-1 & 3 & 2&1&0&0\\1&-2& -3&0&1&0\\ 2& -4&-2&0&0&1 \end{bmatrix}$

$\sim \ \begin{bmatrix}-1 & 3 & 2&1&0&0\\ 0&1& -1&1&1&0\\ 0& 2&2&2&0&1 \end{bmatrix}$

$\sim \ \begin{bmatrix}1 & -3 & -2&-1&0&0\\ 0&1& -1&1&1&0\\ 0& 0&4&0&-2&1 \end{bmatrix}$

$\sim \ \begin{bmatrix}1 & -3 & 0&-1&-1&\frac{1}{2}\\ 0&1& 0&1&\frac{1}{2}&\frac{1}{4}\\ 0& 0&1&0&-\frac{1}{2}&1\end{bmatrix}$

$\sim \ \begin{bmatrix} 1 & 0 & 0&2&\frac{1}{2}&\frac{5}{4}\\ 0&1& 0&1&\frac{1}{2}&\frac{1}{4}\\ 0& 0&1&0&-\frac{1}{2}&1 \end{bmatrix}.$

Hence $\ A^{-1} = \begin{bmatrix} 2&\frac{1}{2}&\frac{5}{4}\\ 1&\frac{1}{2}&\frac{1}{4}\\ 0&-\frac{1}{2}&1 \end{bmatrix}$

TSSFL TECHNOLOGY STACK

Matrix inverse by Elementary Row Operations

Matrix inverse by Elementary Row Operations