Matrix Calculator

Perform matrix operations including addition, subtraction, multiplication, transpose, determinant, and inverse.

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Rows

Cols

Scalar k

Matrix A

Matrix B

Perform addition, subtraction, multiplication, scalar multiplication, transpose, determinant, and inverse operations on matrices up to 4×4 (determinant and inverse are supported for 2×2 and 3×3).

How It Works

How Matrix Calculator Works

Choose your matrix size (2×2 up to 4×4) and enter values into Matrix A and, for most operations, Matrix B — both start out as identity matrices by default. Selecting a tab (A+B, A−B, A×B, k×A, Transpose, det(A), A⁻¹) switches which operation runs when you press Calculate.

Addition and subtraction combine matching entries and require A and B to have identical dimensions. Multiplication (A×B) computes each output entry as the dot product of a row of A with a column of B, which requires A's column count to equal B's row count. Scalar multiplication (k×A) multiplies every entry by k, and Transpose flips rows into columns.

Determinant uses the direct ad−bc formula for 2×2 matrices and cofactor expansion along the first row for 3×3 matrices; the inverse is built from the same cofactor (adjugate) matrix divided by the determinant. Both are limited to 2×2 and 3×3 since larger matrices need a full elimination algorithm instead of a closed-form formula.

Worked Example

See It In Action

For A = [[1,2],[3,4]] and B = [[5,6],[7,8]]: A + B = [[6,8],[10,12]], adding matching entries directly. Multiplying A × B instead uses row-by-column dot products: the top-left entry is (1×5 + 2×7) = 19, giving the full product [[19,22],[43,50]] — a very different result from element-wise addition.

Real-World Use Cases

Who Uses Matrix Calculator and Why

Running several different matrix operations (add, subtract, multiply, scalar, transpose, determinant, inverse) on the same pair of matrices without switching tools.
Quickly checking a 2×2 or 3×3 homework answer for addition, multiplication, or inversion in one place.
Exploring how scalar multiplication (k×A) differs from full matrix multiplication (A×B) side by side.
Getting the determinant or inverse of a small coefficient matrix before moving on to a dedicated linear-system solver.

Common Mistakes

Mistakes to Avoid

Trying to compute determinant or inverse on a 4×4 matrix — this calculator's closed-form formulas only cover 2×2 and 3×3; larger sizes need the elimination-based Matrix Rank or Inverse Matrix Method calculators.
Entering mismatched dimensions for A + B or A − B, which requires both matrices to be exactly the same size.
Confusing scalar multiplication (k×A, multiplying every entry by one plain number) with matrix multiplication (A×B, which uses row-by-column dot products) — they produce very different results.

Pro Tips

Tips for Best Results

Use this calculator for quick, small (up to 3×3 for det/inverse) exploratory checks, then move to the dedicated Matrix Rank or Inverse Matrix Method calculators for anything 4×4 or larger.
Since A and B both default to identity matrices, only the cells you actually change need attention when setting up a quick example.

Troubleshooting

Fixing Common Problems

The determinant or inverse tab isn't giving a result for my matrix. — Confirm your matrix is 2×2 or 3×3 — det(A) and A⁻¹ aren't supported for 4×4 here; use the dedicated Determinant or Inverse Matrix calculator (or Matrix Rank / Inverse Matrix Method) for larger sizes.

A × B says it's undefined. — Check that the number of columns in A matches the number of rows in B — that compatibility is required for any matrix multiplication.

Glossary

Terms Explained

Identity matrix: A square matrix with 1s on the main diagonal and 0s elsewhere, which acts as the "do-nothing" matrix under multiplication.

Scalar multiplication: Multiplying every single entry of a matrix by the same plain number.

FAQ

Frequently Asked Questions

Why are determinant and inverse only available for 2×2 and 3×3 matrices?

Those sizes have simple closed-form formulas (ad−bc for 2×2, cofactor expansion for 3×3). Larger matrices need a general elimination-based algorithm — see the Matrix Rank and Inverse Matrix Method calculators, which handle bigger systems.

What do two matrices need in common to be multiplied?

The number of columns in the first matrix must equal the number of rows in the second — an m×n matrix can only be multiplied by an n×p matrix, producing an m×p result.

How is scalar multiplication different from matrix multiplication?

Scalar multiplication (k×A) multiplies every single entry of A by the same plain number k, while matrix multiplication (A×B) combines rows of one matrix with columns of another using dot products — a fundamentally different operation.

Why do addition and subtraction require matching dimensions?

Since both operations combine entries position by position (A[i][j] ± B[i][j]), every position in A must have a corresponding position in B — matrices of different shapes simply don't line up.