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Matrix Calculator Addition · Multiplication · Determinant · Inverse.

Add, subtract, multiply 2x2 and 3x3 matrices and calculate the determinant and inverse

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Matrix Calculator

Addition, subtraction, multiplication, determinant and inverse for 2x2 and 3x3 matrices

Matrix A

Matrix B

A + B

681012

A − B

-4-4-4-4

A × B (row by column)

19224350

Determinants

det(A) = -2

det(B) = -2

Inverse A⁻¹

-211.5-0.5

Inverse B⁻¹

-433.5-2.5

Matrix calculations – the basics

Addition, multiplication, determinant and inverse for 2x2 and 3x3 matrices

A matrix is a rectangular array of numbers, arranged in rows and columns. This calculator supports square matrices of size 2x2 and 3x3 – the typical scope covered in school and introductory linear algebra courses.

For addition and subtraction, matrices are combined element by element: (A ± B)ᵢⱼ = aᵢⱼ ± bᵢⱼ. Both matrices must have the same size for this.

The matrix multiplication follows the row-by-column rule: cᵢₖ = Σⱼ aᵢⱼ · bⱼₖ. The number of columns of A must equal the number of rows of B — for two square matrices of different sizes (e.g. 2x2 and 3x3), multiplication is therefore not defined.

The determinant of a 2x2 matrix is calculated as ad − bc. For 3x3 matrices, the rule of Sarrus is used, named after the mathematician Pierre Frédéric Sarrus. The determinant decides whether a matrix has an inverse: only if det(A) ≠ 0 does A⁻¹ exist.

Calculation examples

Addition: A + B (2x2)

Addition: A + B (2x2)
PositionBetrag
Matrix A[[1,2],[3,4]]
Matrix B[[5,6],[7,8]]
A + B[[6,8],[10,12]]

Determinant (2x2): A = [[4,3],[6,8]]

Determinant (2x2): A = [[4,3],[6,8]]
PositionBetrag
Formulaa·d − b·c
Substituting4·8 − 3·6
det(A)14

Inverse (2x2): A = [[4,7],[2,6]]

Inverse (2x2): A = [[4,7],[2,6]]
PositionBetrag
det(A) = 4·6 − 7·210
Formula1/det(A) · [[d,−b],[−c,a]]
A⁻¹[[0.6,−0.7],[−0.2,0.4]]

Frequently asked questions about the matrix calculator

Addition, multiplication, determinant and inverse explained clearly

Matrices are added element by element: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ. Both matrices must have the same size (e.g. both 2x2 or both 3x3). Example: [[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]].

Following the row-by-column rule, each element cᵢₖ of the product is calculated as the sum of the products of the i-th row of A and the k-th column of B: cᵢₖ = Σⱼ aᵢⱼ · bⱼₖ. Requirement: the number of columns of A must equal the number of rows of B. For two square matrices of the same size (e.g. both 2x2), multiplication is always possible; for different sizes (e.g. 2x2 times 3x3), it is not.

For a 2x2 matrix [[a,b],[c,d]]: det(A) = a·d − b·c. For a 3x3 matrix, the rule of Sarrus is used: det(A) = a₁₁a₂₂a₃₃ + a₁₂a₂₃a₃₁ + a₁₃a₂₁a₃₂ − a₁₃a₂₂a₃₁ − a₁₁a₂₃a₃₂ − a₁₂a₂₁a₃₃. The determinant indicates whether a matrix is invertible (det ≠ 0).

A matrix is not invertible (singular) exactly when its determinant is 0. In that case, no matrix A⁻¹ exists with A · A⁻¹ = identity matrix. Our calculator detects this case automatically and displays a notice.

For A = [[a,b],[c,d]], A⁻¹ = 1/det(A) · [[d,−b],[−c,a]]. You swap the main diagonal, negate the secondary diagonal, and divide each entry by the determinant ad − bc.

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