What is a matrix determinant?

The determinant is a scalar value derived from a square matrix. It tells you if the matrix is invertible (det ≠ 0), and represents the scaling factor of the linear transformation the matrix represents.

When does a matrix have no inverse?

A matrix has no inverse (is singular) when its determinant is zero. This means the matrix collapses space to a lower dimension and cannot be reversed.

What is the transpose of a matrix?

The transpose flips a matrix along its diagonal — rows become columns and columns become rows. If A is m×n, then Aᵀ is n×m.

Is matrix multiplication commutative?

No. In general, A×B ≠ B×A for matrices. Matrix multiplication is associative and distributive but not commutative.

What size matrices does this calculator support?

This calculator works with 3×3 matrices for all operations. 3×3 covers a wide range of practical problems including 3D transformations and systems of three linear equations.

Math

↵ to calculate

Matrix Calculator — Add Multiply Determinant Inverse

Add, subtract, multiply matrices and find the determinant, transpose, and inverse of 2×2 and 3×3 matrices.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

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📌 Direct Answer & Summary

A matrix calculator performs common matrix operations on 3×3 matrices: determinant, inverse, transpose, addition, and multiplication. Matrices are rectangular arrays of numbers used in linear algebra, computer graphics, physics simulations, machine learning, and engineering.

Matrix A

What is Matrix Calculator — Add Multiply Determinant Inverse?

How to use it

1️⃣ Select the operation using the tabs at the top: Det, Inv, Transpose, Add, or Multiply.
2️⃣ Fill in the values of Matrix A (and Matrix B if required for Add or Multiply).
3️⃣ Click the Calculate button to see the result.
4️⃣ For the determinant, a single scalar is shown; for other operations, the result matrix is displayed.

Formula

Determinant (3×3): expansion along first row using 2×2 minors. Inverse: A⁻¹ = (1/det(A)) × adjugate(A). Transpose: swap rows and columns. Matrix multiply: (AB)ᵢⱼ = Σ AᵢₖBₖⱼ.

💡 See it in action — a real example

For the identity matrix, det = 1 and the inverse equals itself. For matrix addition, each element is added position by position. Multiplying a 3×3 by a 3×3 produces another 3×3 via dot products of rows and columns.

❓ Common questions

What is a matrix determinant?: The determinant is a scalar value derived from a square matrix. It tells you if the matrix is invertible (det ≠ 0), and represents the scaling factor of the linear transformation the matrix represents.
When does a matrix have no inverse?: A matrix has no inverse (is singular) when its determinant is zero. This means the matrix collapses space to a lower dimension and cannot be reversed.
What is the transpose of a matrix?: The transpose flips a matrix along its diagonal — rows become columns and columns become rows. If A is m×n, then Aᵀ is n×m.
Is matrix multiplication commutative?: No. In general, A×B ≠ B×A for matrices. Matrix multiplication is associative and distributive but not commutative.
What size matrices does this calculator support?: This calculator works with 3×3 matrices for all operations. 3×3 covers a wide range of practical problems including 3D transformations and systems of three linear equations.

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