# Find the Inverse of a Matrix Using Row Reduction - Calculator

## Row Reduction to Find the Inverse of a Matrix

An online calculator that calculates the inverse of a square matrix using row reduction is presented.If the matrix \( A^{-1} \) is the inverse of an \( n \times n \) matrix \( A \) , then we have \[ A A^{-1} = I_n \] where \( I_n \) is the \( n \times n \) identity matrix

To find the inverse \( A^{-1} \), we start with the augmented matrix \( [ A | I_n ] \) and then row reduce it. If matrix \( A \) is invertible, the row reduction will end with an augmented matrix in the form

\[ [ I_n | A^{-1} ] \] where the inverse \( A^{-1} \) is the \( n \times n \) on the right side of \( [ I_n | A^{-1} ] \)

NOTE

If while row reducing the augmented matrix, one column or one row of the matrix on the left has zeros only, there no need to continue because the denominator of matrix matrix \( A \) is equal to zero and the matrix is not invertible.

## How to Use the Calculator

Enter the number of columns (and rows) \( n \) below, click on "Generate Matrix" to generate a matrix with random values of its elements. You may change the values of the elements by entring new values and click on "Update Matrix". You may enter the values of the elements of the matrix as integers, decimal numbers such as 1.2 or fractions such as -4/5.The steps per column are shown: In blue the row echelon form and in red the row reduced form.

Enter Number of columns (and rows) \( n = \)

Click here to enter \( n \) and generate a matrix whose elemenst have random values

Change values of coefficients in above matrix (if needed) and click

## More References and links

- Row Reduce Agmented Matrices - Calculator
- Linear Algebra Calculators
- Linear Algebra - Questions with Solutions
- Linear Algebra and its Applications - 5 th Edition - David C. Lay , Steven R. Lay , Judi J. McDonald
- Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres