Find the inverse of any invertible square matrix (up to 6×6) using Gauss-Jordan elimination — every row operation shown step by step.
Matrix Setup
Result
Step-by-Step: Gauss-Jordan Elimination
Method: Gauss-Jordan Elimination
How it works: Form the augmented matrix $[A \mid I]$ and apply row operations to reduce A to the identity matrix I. The right half becomes $A^{-1}$.
Three row operations allowed:
1. $R_i \leftrightarrow R_j$ — Swap two rows
2. $R_i \leftarrow k \cdot R_i$ — Multiply a row by scalar $k \neq 0$
3. $R_i \leftarrow R_i + k \cdot R_j$ — Add a multiple of one row to another
Key fact: $A^{-1}$ exists if and only if $\det(A) \neq 0$.
Matrix inverse: how it is found
The calculator uses Gauss-Jordan elimination. Write the augmented matrix [A | I] and apply row operations until the left half becomes the identity; the right half is then A⁻¹. An inverse exists only for a square matrix with non-zero determinant. For a 2×2 matrix [[a, b],[c, d]] the inverse is 1/(ad − bc) · [[d, −b],[−c, a]].
Common questions
When does a matrix have no inverse?
When it is not square or its determinant is zero (a singular matrix).