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# Modular inverse of a matrix

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

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

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

#### Modular inverse of a matrix

Modular inverse of a matrix

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 an adjugate matrix to find the inverse, which is inefficient for large matrices due to its recursion, but perfectly suits us. The 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 between this calculator and calculator Inverse matrix calculator is modular arithmetic. Modulo operation is used in all calculations, and division by determinant is replaced with multiplication by the modular multiplicative inverse of determinant, refer to Modular Multiplicative Inverse.

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PLANETCALC, Modular inverse of a matrix