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Linear algebra · Matrices

Matrix Operations Calculator

Calculate determinants, inverses, and matrix products for 2×2 and 3×3 matrices with dimension checks and a concise explanation of each operation.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Square real matrices limited to order two or three

Product

(AB)ᵢⱼ = Σaᵢₖbₖⱼ

Inverse condition

AA⁻¹ = Idet(A) ≠ 0

2×2 determinant

det(A) = ad − bc
Vectors & matrices

Operate on small square matrices

Matrix A

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

det(A)

Select an operation and fill every visible matrix cell.

Using the model

Symbols, assumptions and limitations

Interpretation notes

Matrices are limited to 2×2 and 3×3 so the entries and explanations remain readable. An inverse exists only when the determinant is non-zero. Decimal results are rounded for display after full-precision elimination.

Worked examples

See the method in practice

01

2×2 determinant

For A = [[1,2],[3,4]], det(A) = 1×4 − 2×3 = −2.

02

Row-by-column product

Multiplying [1,2] by the column [3,4] gives 1×3 + 2×4 = 11.

Questions

Frequently asked

Why can a matrix lack an inverse?

A square matrix is invertible exactly when its determinant is non-zero.

Why is matrix multiplication not entrywise?

Each output entry is the scalar product of one row from the first matrix and one column from the second.

Are larger matrices supported?

Not in this educational interface; the 2×2 and 3×3 limit keeps steps and mobile layout legible.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Precalculus — Matrices and Matrix Operations

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology