1. What is a Matrix Inverse?

The inverse of a matrix A is another matrix, denoted as A⁻¹, such that when multiplied by the original matrix yields 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:

• \( \det(A) \) — Determinant of matrix A
• \( \text{adj}(A) \) — Adjugate (transpose of cofactor matrix) of A

Explanation: For 2x2 matrices, the calculation is straightforward. For 3x3 matrices, it involves calculating minors, cofactors, and the adjugate matrix.

3. Importance of Matrix Inversion

Details: Matrix inversion is crucial in solving systems of linear equations, computer graphics, cryptography, and many areas of engineering and physics.

4. Using the Calculator

Tips: Select matrix size (2x2 or 3x3), enter all matrix elements. The calculator will compute the determinant and, if possible, the inverse matrix.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is my matrix not invertible?
A: If the determinant is zero, the matrix is singular and has no inverse. This happens when rows/columns are linearly dependent.

Q3: What's 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.

Q4: Are there other methods to find inverses?
A: Yes, including Gaussian elimination and LU decomposition, especially for larger matrices.

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