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# Modular inverse of a matrix

This calculator finds modular inverse of a matrix using adjugate matrix and modular multiplicative inverse

Previous matrix calculators: Determinant of a matrix, Matrix Transpose, Matrix Multiplication, Inverse of a matrix

This calculator finds modular inverse of a matrix using adjugate matrix and modular multiplicative inverse. The theory, as usual, is below the calculator

### Modular inverse of a matrix

Modular inverse of a matrix

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In linear algebra an n-by-n (square) matrix A is called invertible if there exists an n-by-n matrix such that

$AA^{-1} = A^{-1}A = E$

This calculator uses adjugate matrix to find the inverse, which is inefficient for large matrices, due to its recursion, but perfectly suits us here. Final formula uses determinant and the transpose of the matrix of cofactors (adjugate matrix):

$A^{-1} = \frac{1}{\det A}\cdot C^*$

Adjugate of a square matrix is the transpose of the cofactor matrix

${C}^{*}= \begin{pmatrix} {A}_{11} & {A}_{21} & \cdots & {A}_{n1} \\ {A}_{12} & {A}_{22} & \cdots & {A}_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ {A}_{1n} & {A}_{2n} & \cdots & {A}_{nn} \\ \end{pmatrix}$

The cofactor of $a_{ij}$ is $A_{ij}$$A_{ij}=(-1)^{i+j}M_{ij}$
where $M_{ij}$ - determinant of a matrix, which is cut down from A by removing row i and column j (first minor).

The main difference of this calculator from calculator Inverse of a matrix is modular arithmetic. Modulo operation is used in all calculations and division by determinant is replaced with multiplication by modular multiplicative inverse of determinant, refer to Modular Multiplicative Inverse.