Matrix calculator

Matrices are rectangular grids of numbers used throughout linear algebra, computer graphics, statistics, machine learning and engineering. This calculator performs the common operations: addition, subtraction, multiplication, transpose, determinant and inverse, so you can check homework, verify code or solve small systems of equations.

How to use it

1. Set the dimensions of matrix A, and of matrix B if the operation needs two.
2. Enter the values.
3. Choose the operation and read the result.

Rules to remember

• Addition and subtraction need matrices of the same size.
• Multiplication A × B needs the number of columns of A to equal the number of rows of B. A 2×3 times a 3×4 gives a 2×4 result. Matrix multiplication is not commutative: A × B is usually different from B × A.
• Determinant and inverse exist only for square matrices, and the inverse exists only when the determinant is not zero.

Worked example

For A = [[4, 7], [2, 6]]:

• Determinant = 4 × 6 - 7 × 2 = 10.
• Inverse = (1 / 10) × [[6, -7], [-2, 4]] = [[0.6, -0.7], [-0.2, 0.4]].
• Transpose = [[4, 2], [7, 6]].

Check: A × A⁻¹ = [[4×0.6 + 7×(-0.2), 4×(-0.7) + 7×0.4], ...] = [[1, 0], [0, 1]], the identity matrix.

Practical uses

• Solving linear systems: Ax = b gives x = A⁻¹b.
• 2D and 3D graphics: rotation, scaling and translation are matrix multiplications.
• Statistics: covariance matrices and linear regression.

Numerical caution

A determinant very close to zero means the matrix is nearly singular, and its inverse will contain huge values that amplify rounding errors.