Matrix Calculator
Matrix Operations Tool
Matrix A
Matrix B
Understanding Matrices
A matrix is simply a rectangular grid of numbers — but that simple structure underlies computer graphics, machine learning, engineering simulations, and systems of linear equations.
These operations combine matrices entry by entry, which is why the two matrices must be exactly the same size — there's no other way to pair up corresponding cells.
Matrix multiplication combines rows of the first matrix with columns of the second, and is the mathematical foundation for transforming coordinates, rotating 3D graphics, and chaining linear systems together.
Determinant & Inverse
2×2 Determinant
Inverse (via Adjugate)
A matrix only has an inverse when its determinant is non-zero. When it's zero, the matrix is singular — analogous to trying to divide by zero — and no inverse exists.
Transpose
Transposing a matrix reflects it across its main diagonal, turning every row into a column. Unlike determinant and inverse, transpose places no restriction on shape — you can transpose a rectangular matrix of any size, and it always produces a valid result.
This calculator supports matrices up to 4 rows by 4 columns.
A determinant of zero means the matrix has no inverse.
Rows become columns, and columns become rows.
Key Takeaways
- Addition and subtraction require matching dimensions, while multiplication only requires the inner dimensions to match.
- A zero determinant means no inverse exists — the matrix is singular, the matrix equivalent of dividing by zero.
- Matrix multiplication is not commutative: the order of A × B matters, and usually gives a different result than B × A.
Frequently Asked Questions
- Choose the Two Matrices tab to add, subtract, or multiply two matrices, or the Single Matrix tab to find a determinant, inverse, or transpose.
- Use the row/column dropdowns to size the grid (up to 4×4), then fill in the cells.
- Pick an operation and click "Calculate" to see the resulting matrix (or number, for a determinant).
Addition and subtraction require both matrices to have exactly the same number of rows and columns. Multiplication requires the number of columns in Matrix A to match the number of rows in Matrix B. Determinant and inverse require a square matrix (same number of rows and columns). Transpose works on a matrix of any shape.
The determinant is a single number computed from a square matrix that reveals important properties about it — most notably, a matrix has an inverse if and only if its determinant is not zero. Geometrically, the absolute value of the determinant represents how much the matrix scales area (2×2) or volume (3×3).
A matrix is called singular when its determinant equals zero, and singular matrices have no inverse — just like you can't divide a number by zero. This typically happens when one row or column can be written as a combination of the others, meaning the matrix collapses information rather than preserving it.
Transposing a matrix flips it over its main diagonal, turning rows into columns and columns into rows. A matrix that started as 2 rows × 3 columns becomes 3 rows × 2 columns after transposing. Unlike determinant and inverse, transpose works on matrices of any shape, not just square ones.
No — that common mistake is called element-wise multiplication, and it's a different operation. Standard matrix multiplication combines each row of the first matrix with each column of the second, summing the products of corresponding entries. This is also why matrix multiplication isn't commutative: A × B usually doesn't equal B × A.