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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
In linear algebra, an n-by-n (square) matrix A is called invertible if there exists an n-by-n matrix such that
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):
Adjugate of a square matrix is the transpose of the cofactor matrix
The cofactor of is
where - 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 Calculator.