Matrix Modulo Calculator — Advanced Operations

Modulus (n > 0)

Operation

For Power, supply non-negative integer k.

Matrix A (rows by newlines, numbers by spaces or commas)

Matrix B (only for +, −, ×)

Modular reduction: for any integer a, a (mod n) is the remainder 0…n−1.
Add/Subtract: (A ± B) (mod n) has entries a_ij ± b_ij reduced modulo n.
Multiply: (A×B)_ij = Σ_k a_ikb_kj, then reduce each entry modulo n.
Power: A^k via repeated squaring with all intermediate entries reduced modulo n.
Determinant: computed as the integer determinant (Bareiss algorithm), then reduced modulo n.
Inverse (mod n): A is invertible modulo n iff gcd(det(A), n) = 1. Inverse is found by Gauss–Jordan elimination over ℤ/nℤ when pivots are invertible modulo n.

A (3×3)

6 24 1
13 16 10
20 17 15

B (3×3)

3 12 17
5 9 2
8 4 11

With modulus 26, A is invertible because gcd(det(A), 26) = 1.

Use one row per line. Separate entries by spaces or commas. All rows must have the same number of entries.

Over ℤ/nℤ, a matrix is invertible only if its determinant is coprime with n. If gcd(det(A), n) ≠ 1, an inverse modulo n does not exist.

Yes. Multiplication requires columns of A equal rows of B. Only powers, determinant, and inverse require square matrices.

All entries are reduced to the range 0…n−1 using a safe remainder function, so negatives wrap correctly under the chosen modulus.

We use the fraction-free Bareiss algorithm to compute the exact integer determinant, and then reduce it modulo n for the final value.

Yes. Use the PDF button; it captures the result table and any step logs shown for inverses, packaging them into a neatly formatted PDF.

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No matrix to display yet, or the chosen operation returned no matrix.

Tip: For composite moduli, invertibility is stricter than in fields. If an inverse fails, try a different modulus or matrix.

Operation	Dimension Rule	Entry / Rule (mod n)	Output Size
A + B	Same rows and columns	c_ij = (a_ij + b_ij) mod n	r × c
A − B	Same rows and columns	c_ij = (a_ij − b_ij) mod n	r × c
A × B	A: r × k, B: k × c	c_ij = Σ_ta_itb_tj mod n	r × c
A^k	A is square n × n	Repeated squaring with reductions mod n	n × n
det(A)	A is square n × n	Integer det (Bareiss), then det(A) mod n	1 × 1 scalar
A⁻¹	A is square n × n	Exists iff gcd(det(A), n) = 1	n × n (if exists)

n (modulus)	Structure	Invertible Condition	Typical Context
26	Composite (2 × 13)	gcd(det(A), 26) = 1	Classical Hill cipher examples
29	Prime (field)	det(A) ≠ 0 (mod 29)	Finite-field arithmetic demos
97	Prime (field)	det(A) ≠ 0 (mod 97)	Number theory experiments
256	Composite (2⁸)	gcd(det(A), 256) = 1 (det odd)	Byte-wise arithmetic settings
12	Composite	gcd(det(A), 12) = 1	Modular tiling / lattice toy cases

What input format do matrices require?