How do I use the Matrix Calculator?

Enter the required values, then calculate. The Matrix Calculator shows results, formulas, and step-by-step work when available. No account is required.

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How it works

Supports rectangular editors, A+B / A−B / AB, cA, Aⁿ, Aᵀ, det, and A⁻¹ (inverse up to 3×3). Matrix multiplication is not commutative.

det 2×2 = ad−bc · c_(i,j)=Σ a_(i,k)b_(k,j)

When to use

Use this tool when you need to: resize rows/cols (1–4). Try det of [[1,2],[3,4]] = −2. Use All 0 / Random tools. Power needs a square matrix.

How to use this tool

Enter your values in the fields above.
Review the formula and any mode options for your problem.
Read the result and the step-by-step solution.

Resize rows/cols (1–4). Try det of [[1,2],[3,4]] = −2. Use All 0 / Random tools. Power needs a square matrix.

Frequently asked questions

What is the Matrix Calculator?: Supports rectangular editors, A+B / A−B / AB, cA, Aⁿ, Aᵀ, det, and A⁻¹ (inverse up to 3×3). Matrix multiplication is not commutative.
How do I use the Matrix Calculator?: Enter the required values, then calculate. The Matrix Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Matrix Calculator use?: det 2×2 = ad−bc · c_(i,j)=Σ a_(i,k)b_(k,j)

Browse categories

Matrix calculator

Edit Matrix A and B (up to 4×4). Run determinant, inverse, transpose, power, scalar multiply, A+B, A−B, or AB — with fill tools and step-by-step work.

How it works

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Dimensions m×n mean m rows and n columns. Element a_(i,j) is the entry in row i and column j. Matrices appear in physics, computer graphics, statistics, and numerical analysis.

Addition and subtraction

Addition and subtraction require equal size. Combine corresponding entries: c_(i,j) = a_(i,j) ± b_(i,j). Example: [[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]. Same matrices subtracted: [[−4,−4],[−4,−4]].

Multiplication, power, and transpose

Scalar: multiply every entry by c. Matrix product AB requires cols(A) = rows(B). Entry c_(i,j) is the dot product of row i of A and column j of B. In general AB ≠ BA. Power Aⁿ is defined for square A only (A multiplied by itself n times). A⁰ = I. Transpose Aᵀ swaps rows and columns (m×n becomes n×m).

Determinant

The determinant is a scalar from a square matrix. For 2×2: |A| = ad − bc. For 3×3, Laplace / Leibniz expansion: a(ei−fh) − b(di−fg) + c(dh−eg). Larger sizes use the same expansion pattern (this tool supports up to 4×4).

Inverse

A⁻¹ satisfies A·A⁻¹ = A⁻¹·A = I (identity). It exists only when det(A) ≠ 0. For 2×2: A⁻¹ = (1/(ad−bc)) · [[d,−b],[−c,a]]. For 3×3 the adjugate (cofactor transpose) scaled by 1/det is used. Identity matrices have 1s on the diagonal and 0s elsewhere.

Related calculators

Scientific expressions and large-integer arithmetic complement matrix algebra for general science and numerical work.

How it works

Supports rectangular editors, A+B / A−B / AB, cA, Aⁿ, Aᵀ, det, and A⁻¹ (inverse up to 3×3). Matrix multiplication is not commutative.

det 2×2 = ad−bc · c_(i,j)=Σ a_(i,k)b_(k,j)

When to use

Use this tool when you need to: resize rows/cols (1–4). Try det of [[1,2],[3,4]] = −2. Use All 0 / Random tools. Power needs a square matrix.

How to use this tool

Enter your values in the fields above.
Review the formula and any mode options for your problem.
Read the result and the step-by-step solution.

Resize rows/cols (1–4). Try det of [[1,2],[3,4]] = −2. Use All 0 / Random tools. Power needs a square matrix.

Frequently asked questions

What is the Matrix Calculator?: Supports rectangular editors, A+B / A−B / AB, cA, Aⁿ, Aᵀ, det, and A⁻¹ (inverse up to 3×3). Matrix multiplication is not commutative.
How do I use the Matrix Calculator?: Enter the required values, then calculate. The Matrix Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Matrix Calculator use?: det 2×2 = ad−bc · c_(i,j)=Σ a_(i,k)b_(k,j)