Step-by-Step Matrix Inverse Calculator

Using Gauss-Jordan Elimination

Calculate the inverse of any square matrix with detailed step-by-step row operations. Perfect for students learning linear algebra.

How Gauss-Jordan Elimination Works - Step by Step:

  1. Step 0: Start with augmented matrix [A | I] where I is the identity matrix
  2. For each column (from left to right):
  3. Pivot Selection: Find a non-zero element in the current column below or at the current row
  4. Row Swap: If needed, swap the current row with the pivot row
  5. Scale Pivot: Multiply the pivot row to make the pivot element equal to 1
  6. Eliminate Others: For all other rows, subtract an appropriate multiple of the pivot row to make their elements in the current column equal to 0
  7. Repeat: Move to the next column and repeat until the left side becomes the identity matrix
  8. Result: The right side of the augmented matrix becomes the inverse A⁻¹

Note: If any column on the left becomes all zeros during elimination, the matrix is not invertible (singular).

Enter Matrix Size

(n × n matrix, where 1 ≤ n ≤ 8)

Step-by-Step Solution

Each step below shows the matrix after one row operation with explanation:

About This Calculator

This calculator uses exact fraction arithmetic to avoid rounding errors. You can enter:

The algorithm shows every single row operation, making it an excellent learning tool for understanding Gauss-Jordan elimination.