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Calculator Answers

Matrix Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A matrix calculator performs fundamental linear algebra operations on two-dimensional arrays of numbers, specifically supporting two-by-two and three-by-three matrices. The tool calculates matrix addition, subtraction, scalar multiplication, matrix multiplication, determinants, transposes, and matrix inverses. Linear algebra students, engineers, and computer graphics programmers use this utility to verify matrix arithmetic, solve transformation matrices, and check computational models without performing tedious manual row and column operations.

Pick a size (2×2 or 3×3) and an operation. Enter your matrices and the calculator returns the result. Determinant and inverse run on Matrix A alone; addition, subtraction, and multiplication use both A and B.

See also: Matrix Multiplication Calculator, Determinant Calculator, and Eigenvector Calculator.

Quick Answer

Perform matrix arithmetic and calculations. Enter your matrix values to compute determinants, transposes, inverses, and products.

Matrix size

Operation

Matrix A

Matrix B

This page handles operations between matrices. To solve a system of equations or compute reduced row echelon form, use the system of equations or RREF calculator instead.

Matrix result

Result A × B

0 0 | 0 0

Row 10 0
Row 20 0

Inverse exists only when det(A) ≠ 0. The 3×3 inverse uses the adjugate method (cofactors transposed, divided by the determinant).

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Examples

[[1,2],[3,4]] + [[5,6],[7,8]]

[[6,8],[10,12]]

[[1,2],[3,4]] × [[2,0],[1,2]]

[[4,4],[10,8]]

det([[1,2],[3,4]])

= −2

inv([[2,0],[0,2]])

[[0.5,0],[0,0.5]]

How it works

Matrix operations are well-defined arithmetic on rectangular arrays of numbers. The calculator applies the standard element-wise rules for addition, subtraction, and scalar multiplication, and the row-by-column rule for matrix multiplication.

Multiplication · (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ

2×2 determinant · det = ad − bc

2×2 inverse · A⁻¹ = (1/det)·[[d,−b],[−c,a]]

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Matrix Multiplication CalculatorMultiply two matrices step by step. Enter the values for matrix A and matrix B to see the product, the dimensions, and how each entry is computed from the row-by-column dot products.Determinant CalculatorCalculate the determinant of a 2x2 or 3x3 matrix instantly with step-by-step minor expansions.Eigenvector CalculatorFind eigenvalues and unit eigenvectors of a 2×2 matrix. Returns trace, determinant, discriminant, both eigenvalues, and the corresponding unit eigenvectors.

Frequently asked questions

Addition (A + B), subtraction (A − B), scalar multiplication (s · A), matrix multiplication (A × B), transpose (Aᵀ), determinant det(A), and inverse A⁻¹. Both 2×2 and 3×3 matrices are supported.

This page intentionally caps at 3×3 to keep the UI tight and the algorithms numerically reliable. For larger matrices and reduced row echelon form, use the RREF calculator.

When its determinant is nonzero. If det(A) = 0, the matrix is singular and has no inverse. The calculator reports the determinant alongside the inverse so you can see why an inverse may or may not exist.

For 2×2, it uses the closed-form swap-and-negate formula. For 3×3, it uses the adjugate method: take the cofactor matrix, transpose it, and divide every entry by the determinant.

No. To solve a linear system Ax = b, use the system of equations calculator (for 2 or 3 variables) or the RREF calculator (which handles the augmented matrix directly).

For addition, subtraction, and multiplication this calculator assumes A and B are the same square size. Add/subtract requires equal dimensions; multiplication for non-square matrices needs the column count of A to match the row count of B, which is out of scope here.