Matrix Inverse Calculator | Invert Matrix Online
Calculate the inverse of a square matrix online.
How to Use
- Enter the elements of a square matrix in the text area.
- Ensure the number of rows matches the number of columns.
- Use spaces or commas to separate numbers in a row, and use new lines for each row.
- Click 'Calculate Inverse' to view the inverse matrix.
About Inverse Matrices
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate), if there exists an n-by-n square matrix B such that AB = BA = I, where I is the n-by-n identity matrix.
Condition for Invertibility
A square matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is called singular or degenerate, meaning it squishes space into a lower dimension, and this process cannot be reversed to uniquely retrieve the original space.
The Identity Matrix
The identity matrix, I, is a square matrix with 1s on the main diagonal and 0s elsewhere. It acts as the multiplicative identity in matrix algebra, analogous to the number 1 in regular multiplication.
Properties of Inverses
- If A is invertible, its inverse is unique.
- (A⁻¹)⁻¹ = A
- (AB)⁻¹ = B⁻¹A⁻¹ (Notice the reversed order)
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ
Applications
- Cryptography: Encrypting and decrypting messages (e.g., Hill cipher).
- Computer Graphics: Reversing transformations like scaling, rotation, and translation.
- Solving linear systems equations of the form Ax = b (x = A⁻¹b).
- Control theory and signal processing.