1. What is an Inverse Matrix?

The inverse of a square matrix A, denoted as A⁻¹, is a matrix that when multiplied by A 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 (adjoint) of matrix A

Explanation: For 2x2 matrices, the adjugate is simply the cofactor matrix. For 3x3 matrices, it's the transpose of the cofactor 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 inverse (if it exists).

5. Frequently Asked Questions (FAQ)

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

Q2: Why is my matrix not invertible?
A: If the determinant is zero, the matrix is singular and has no inverse.

Q3: What's the difference between adjugate and transpose?
A: Adjugate is the transpose of the cofactor matrix, not just the transpose of the original matrix.

Q4: Are there other methods to find inverses?
A: Yes, Gaussian elimination and LU decomposition are alternative methods.

Q5: What's the practical limit for matrix inversion?
A: While mathematically possible for any invertible matrix, computational limitations make large matrices (1000×1000+) impractical for exact inversion.