Matrix Calculator

Math

Perform matrix operations including Addition, Subtraction, Multiplication, Determinant, Transpose, and Inverse for 2×2 and 3×3 matrices.

Matrix Multiplication A × B

Matrix Calculation Steps

Row × Column Dot Product

C[i,j] = Σ A[i,k] × B[k,j]

Compute the dot product of row i of matrix A with column j of matrix B.

How This Calculator Works

A matrix is a two-dimensional rectangular array of numbers arranged into rows and columns. Matrix algebra provides the mathematical framework for solving simultaneous linear equations, transforming 2D/3D graphics, running quantum mechanics simulations, and training machine learning algorithms.

📐 Matrix Multiplication & Determinant Formulas

C[i,j] = Σ (A[i,k] · B[k,j]) | det([[a,b],[c,d]]) = a·d - b·c

A[i,k]: Element in row i and column k of matrix A

B[k,j]: Element in row k and column j of matrix B

det(M): Determinant scalar indicating geometric scaling factor

M⁻¹: Inverse matrix such that M · M⁻¹ = I (Identity)

Key Mathematical Rules & Properties

Matrix Multiplication Order Matters

Matrix multiplication is non-commutative: A × B is generally NOT equal to B × A.

Singular Matrix Condition

A square matrix has an inverse if and only if its determinant is non-zero (det(A) ≠ 0). If det(A) = 0, the matrix is singular.

Transpose Operation

Transposing a matrix swaps its rows and columns: (Aᵀ)[i,j] = A[j,i].

Scalar Multiplication

Multiplying a matrix by a scalar k multiplies every entry in the matrix by k.

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Worked Example: 2×2 Matrix Multiplication

Multiply Matrix A = [[1, 2], [3, 4]] by Matrix B = [[2, 0], [1, 3]].

Row 1 · Col 1: (1·2) + (2·1) = 2 + 2 = 4
Row 1 · Col 2: (1·0) + (2·3) = 0 + 6 = 6
Row 2 · Col 1: (3·2) + (4·1) = 6 + 4 = 10
Row 2 · Col 2: (3·0) + (4·3) = 0 + 12 = 12

Final Answer: [[4, 6], [10, 12]]

Frequently Asked Questions

Why can we not divide matrices directly?

Matrix division is not defined in standard linear algebra. Instead, we multiply by the multiplicative inverse: A ÷ B is computed as A × B⁻¹.

What is the Identity matrix (I)?

The identity matrix has 1s on the main diagonal and 0s elsewhere. Multiplying any compatible matrix by I leaves it unchanged (A · I = A).

What does the determinant represent geometrically?

For a 2×2 matrix, the absolute value of the determinant equals the area of the parallelogram formed by its column vectors. For 3×3, it equals the volume of the parallelepiped.

Can a non-square matrix have an inverse?

Standard two-sided inverses only exist for square matrices with non-zero determinants. Rectangular matrices may have pseudo-inverses (Moore-Penrose inverse).

Understanding your calculation

Method, example and limitations

How the calculation works

Matrix operations follow the selected dimensions: addition combines matching entries, multiplication uses row-by-column products, and inversion requires an invertible square matrix.

Example to check

Multiplying the 2-by-2 identity matrix by any compatible 2-row matrix leaves that matrix unchanged.

Before using the result

Matrix multiplication is generally not commutative. A zero determinant prevents inversion, and nearly singular matrices can amplify rounding errors.

Calculation methodology · Report a result that needs review