Matrix Calculator
Add, subtract, multiply, transpose, power, determinant and inverse — for any matrices up to 6×6.
What is a matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns is called an m×n matrix; the entry in row i and column j is written aᵢⱼ. Matrices are the workhorses of linear algebra and show up anywhere a system of linear relationships needs to be represented compactly.
They power computer graphics (every 3D rotation, scaling and projection is a matrix multiplication), statistics (design matrices in regression), physics (states in quantum mechanics, stress tensors in mechanics), economics (input–output models) and machine learning (essentially every modern neural network is a chain of matrix operations).
Matrix calculator, piece by piece
Each card explains one thing this tool actually does with your grid — how it combines two matrices, how it produces a single number (determinant) from a square one, and how it inverts a matrix when that number isn't zero.
Add, subtract, scalar × — the shape-preserving operations
Addition and subtraction work entry by entry, so both matrices must have the same rows and columns. Scalar multiplication is even simpler — every entry is multiplied by the scalar. The output keeps the shape of the input. If your two matrices don't share dimensions, the calculator refuses instead of guessing.
Multiply, power, transpose — how shapes rearrange
Matrix × matrix requires the inner dimensions to match: an m×n times an n×p produces an m×p, and each output entry is the dot product of a row of A with a column of B. Raising a matrix to a positive integer power only works for squares (repeated self-multiply). The transpose just flips rows into columns, so m×n becomes n×m. Multiplication is not commutative — A·B and B·A are usually different even when both exist.
Determinant & inverse — the square-matrix helpers
The determinant collapses a square matrix into a single number that says how much the matrix scales area or volume; the sign tells you if it flips orientation. For 2×2 that's simply ad − bc, for larger sizes the calculator uses LU decomposition to stay fast at 5×5 and 6×6. Once det is known the inverse follows: A⁻¹ = (1/det)·adj(A), which the tool computes with Gauss–Jordan. If det(A) = 0 the matrix is singular and the calculator reports that rather than returning nonsense.
Common mistakes
- Trying to add matrices of different sizes. Addition is entry by entry, so the shapes must match exactly.
- Multiplying matrices in the wrong order. Even when both products are defined, A × B and B × A are usually different.
- Multiplying two matrices whose inner dimensions do not agree — for example a 2×3 times a 2×2. Check that cols(A) = rows(B).
- Assuming every square matrix has an inverse. Singular matrices (det = 0) do not; the calculator will flag this.
- Confusing the transpose with the inverse. Aᵀ is defined for every matrix; A⁻¹ is not.
Features of this calculator
- Two independent input grids with adjustable rows and columns up to 6×6
- Fill helpers: all zeros, all ones and random small integers
- Per-matrix unary tools: transpose, square (A²), determinant and inverse
- Combined operations: A + B, A − B, A × B and swap A ↔ B
- Copy any result matrix straight back into A or B for chained work
- Clear error messages for singular matrices and dimension mismatches
Frequently asked questions
+When can two matrices be added?
Only when they have identical dimensions. Addition is done position by position.
+When can two matrices be multiplied?
When the number of columns of the first matrix matches the number of rows of the second. An m×n times an n×p produces an m×p matrix.
+Which matrices have an inverse?
Only square matrices whose determinant is non-zero. If det(A) = 0 the matrix is singular and there is no inverse.
+Is matrix multiplication commutative?
No. A × B is generally different from B × A, even when both products are defined and the same shape.
+What is the transpose used for?
It is the natural companion of the dot product, appears throughout statistics (Aᵀ A shows up in least squares), and turns row-based data into column-based data.
+Why does the calculator cap dimensions at 6×6?
Above 6×6 the input grid becomes unwieldy on screen. All the underlying algorithms scale much further; the limit is a UX choice.