Inverse of [5211]\left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]

The calculator will find the inverse of the square 22x22 matrix [5211]\left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right], with steps shown.

Related calculators: Gauss-Jordan Elimination Calculator, Pseudoinverse Calculator

A

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Calculate [5211]1\left[\begin{array}{cc}5 & 2\\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:

[52101101]\left[\begin{array}{cc|cc}5 & 2 & 1 & 0\\1 & 1 & 0 & 1\end{array}\right]

Divide row 11 by 55: R1=R15R_{1} = \frac{R_{1}}{5}.

[1251501101]\left[\begin{array}{cc|cc}1 & \frac{2}{5} & \frac{1}{5} & 0\\1 & 1 & 0 & 1\end{array}\right]

Subtract row 11 from row 22: R2=R2R1R_{2} = R_{2} - R_{1}.

[125150035151]\left[\begin{array}{cc|cc}1 & \frac{2}{5} & \frac{1}{5} & 0\\0 & \frac{3}{5} & - \frac{1}{5} & 1\end{array}\right]

Multiply row 22 by 53\frac{5}{3}: R2=5R23R_{2} = \frac{5 R_{2}}{3}.

[125150011353]\left[\begin{array}{cc|cc}1 & \frac{2}{5} & \frac{1}{5} & 0\\0 & 1 & - \frac{1}{3} & \frac{5}{3}\end{array}\right]

Subtract row 22 multiplied by 25\frac{2}{5} from row 11: R1=R12R25R_{1} = R_{1} - \frac{2 R_{2}}{5}.

[101323011353]\left[\begin{array}{cc|cc}1 & 0 & \frac{1}{3} & - \frac{2}{3}\\0 & 1 & - \frac{1}{3} & \frac{5}{3}\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 [13231353][0.3333333333333330.6666666666666670.3333333333333331.666666666666667].\left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right]\approx \left[\begin{array}{cc}0.333333333333333 & -0.666666666666667\\-0.333333333333333 & 1.666666666666667\end{array}\right].A