1. What is Matrix Inverse?

The inverse of a matrix A is another matrix A⁻¹ such that when they are multiplied together, the result is the identity matrix. Not all matrices have inverses - only square matrices with non-zero determinants are invertible.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( A^{-1} \) — Inverse matrix
• \( adj(A) \) — Adjugate (adjoint) of matrix A
• \( det(A) \) — Determinant of matrix A

Explanation: The inverse is calculated by dividing the adjugate matrix by the determinant. The adjugate is the transpose of the cofactor matrix.

3. Importance of Matrix Inverse

Details: Matrix inverses are fundamental in solving systems of linear equations, computer graphics, cryptography, and many areas of engineering and physics.

4. Using the Calculator

Tips: Enter your square matrix with elements separated by commas and rows separated by new lines. The matrix must be square (same number of rows and columns) and have a non-zero determinant.

5. Frequently Asked Questions (FAQ)

Q1: What matrices have inverses?
A: Only square matrices (n×n) with non-zero determinants are invertible. These are called non-singular or invertible matrices.

Q2: What is the identity matrix?
A: A square matrix with 1s on the diagonal and 0s elsewhere. Multiplying any matrix by its inverse gives the identity matrix.

Q3: How is the adjugate matrix calculated?
A: The adjugate is the transpose of the cofactor matrix, where each element is replaced by its cofactor (signed minor).

Q4: Are there other methods to find matrix inverses?
A: Yes, methods include Gaussian elimination, LU decomposition, and iterative methods for large matrices.

Q5: What are practical applications of matrix inverses?
A: Used in solving linear systems, computer graphics transformations, least squares regression, and encryption algorithms.