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Inverse of $$$\left[\begin{array}{ccc}1 & 1 & 1\\2 & 3 & 4\\3 & 1 & 1\end{array}\right]$$$

The calculator will find the inverse of the square $$$3$$$x$$$3$$$ matrix $$$\left[\begin{array}{ccc}1 & 1 & 1\\2 & 3 & 4\\3 & 1 & 1\end{array}\right]$$$, with steps shown.

Related calculators: Gauss-Jordan Elimination Calculator, Pseudoinverse Calculator

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Your Input

Calculate $$$\left[\begin{array}{ccc}1 & 1 & 1\\2 & 3 & 4\\3 & 1 & 1\end{array}\right]^{-1}$$$ using the Gauss-Jordan elimination.

Solution

To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be the inverse matrix.

So, augment the matrix with the identity matrix:

$$$\left[\begin{array}{ccc|ccc}1 & 1 & 1 & 1 & 0 & 0\\2 & 3 & 4 & 0 & 1 & 0\\3 & 1 & 1 & 0 & 0 & 1\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$2$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 2 R_{1}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 1 & 1 & 1 & 0 & 0\\0 & 1 & 2 & -2 & 1 & 0\\3 & 1 & 1 & 0 & 0 & 1\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$3$$$ from row $$$3$$$: $$$R_{3} = R_{3} - 3 R_{1}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 1 & 1 & 1 & 0 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & -2 & -2 & -3 & 0 & 1\end{array}\right]$$$

Subtract row $$$2$$$ from row $$$1$$$: $$$R_{1} = R_{1} - R_{2}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 0 & -1 & 3 & -1 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & -2 & -2 & -3 & 0 & 1\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$2$$$ to row $$$3$$$: $$$R_{3} = R_{3} + 2 R_{2}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 0 & -1 & 3 & -1 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & 0 & 2 & -7 & 2 & 1\end{array}\right]$$$

Divide row $$$3$$$ by $$$2$$$: $$$R_{3} = \frac{R_{3}}{2}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 0 & -1 & 3 & -1 & 0\\0 & 1 & 2 & -2 & 1 & 0\\0 & 0 & 1 & - \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right]$$$

Add row $$$3$$$ to row $$$1$$$: $$$R_{1} = R_{1} + R_{3}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & - \frac{1}{2} & 0 & \frac{1}{2}\\0 & 1 & 2 & -2 & 1 & 0\\0 & 0 & 1 & - \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right]$$$

Subtract row $$$3$$$ multiplied by $$$2$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 2 R_{3}$$$.

$$$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & - \frac{1}{2} & 0 & \frac{1}{2}\\0 & 1 & 0 & 5 & -1 & -1\\0 & 0 & 1 & - \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right]$$$

We are done. On the left is the identity matrix. On the right is the inverse matrix.

Answer

The inverse matrix is $$$\left[\begin{array}{ccc}- \frac{1}{2} & 0 & \frac{1}{2}\\5 & -1 & -1\\- \frac{7}{2} & 1 & \frac{1}{2}\end{array}\right] = \left[\begin{array}{ccc}-0.5 & 0 & 0.5\\5 & -1 & -1\\-3.5 & 1 & 0.5\end{array}\right].$$$A