🔢 Inverse Matrix Calculator
Calculate the inverse of square matrices with detailed step-by-step solutions. Supports 2×2 to 5×5 matrices using the Gauss-Jordan elimination method.
Frequently Asked Questions
The inverse of a matrix A is another matrix A⁻¹ such that when multiplied together, they produce the identity matrix: A × A⁻¹ = A⁻¹ × A = I. It's similar to the reciprocal of a number.
A matrix inverse doesn't exist when the matrix is singular, meaning its determinant equals zero. This happens when the matrix rows are linearly dependent or the matrix is not full rank.
Gauss-Jordan elimination is a method to find the matrix inverse by creating an augmented matrix [A|I] and performing row operations until the left side becomes the identity matrix. The right side then becomes the inverse.
No, only square matrices (same number of rows and columns) can have an inverse. For non-square matrices, you might want to calculate the pseudoinverse instead.
The determinant is a scalar value that indicates if a matrix is invertible (non-zero determinant) or singular (zero determinant). The inverse is an entire matrix that "undoes" the original matrix when multiplied.
Our calculator uses high-precision floating-point arithmetic with configurable decimal places. For exact results with fractions, enable the fraction output mode.
Matrix inverses are used in solving systems of linear equations, computer graphics transformations, engineering calculations, statistics, and many areas of mathematics and science.
Larger matrices require more computational steps. The time complexity increases significantly with matrix size. For matrices larger than 5x5, consider using specialized mathematical software.
Disclaimer
This calculator is for educational and learning purposes only. Results are provided for reference and should not be used for critical engineering or scientific calculations. For important applications, please verify results using professional mathematical software.