Matrix Inverse Modulo Calculator

Matrix Inverse Modulo Formula:

\[ \text{inv\_mod} = (\text{inv} \times \text{det\_mod\_inverse}) \mod m \]

Inverse Matrix Modulo:

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1. What is Matrix Inverse Modulo?

The matrix inverse modulo operation finds a matrix that, when multiplied with the original matrix modulo m, yields the identity matrix. This is crucial in cryptography and linear algebra over finite fields.

2. How Does the Calculator Work?

The calculator uses the following steps:

\[ \text{inv\_mod} = (\text{adjugate(matrix)} \times \text{det\_mod\_inverse}) \mod m \]

Where:

\( \text{adjugate(matrix)} \) — Transpose of the cofactor matrix
\( \text{det\_mod\_inverse} \) — Modular inverse of the determinant
\( m \) — Modulus value

Explanation: The matrix must be square and its determinant must be coprime with the modulus (have a modular inverse).

3. Importance of Modular Inverse

Details: Matrix inverses modulo m are essential in cryptographic algorithms like Hill cipher, error-correcting codes, and solving systems of linear congruences.

4. Using the Calculator

Tips: Enter matrix elements separated by commas, rows separated by semicolons. The modulus must be a positive integer. The matrix must be square and invertible modulo m.

5. Frequently Asked Questions (FAQ)

Q1: When does a matrix have an inverse modulo m?
A: When the determinant and modulus are coprime (GCD(det, m) = 1).

Q2: What's the difference between regular and modular inverse?
A: Modular inverse works within a finite field (mod m) rather than real numbers.

Q3: Can any matrix have a modular inverse?
A: Only square matrices with determinant coprime to the modulus have inverses.

Q4: What are common applications?
A: Cryptography, coding theory, and solving linear equations in finite fields.

Q5: Why might the calculator fail?
A: If the matrix isn't square, modulus isn't positive, or determinant has no inverse modulo m.