How does matrix multiplication work?

Each entry of the product is the dot product of a row from the first matrix with a column from the second. An m×n matrix times an n×p matrix gives an m×p result — the inner dimensions must match or the product is undefined.

Why isn't matrix multiplication commutative?

In general AB ≠ BA: the row-by-column structure means order matters, and reversing the order can even change the result's dimensions or make the product undefined. Always keep matrices in the intended order when setting up a calculation.

When can I add or subtract two matrices?

Only when they have identical dimensions. Addition and subtraction are element-wise: add or subtract each entry with the corresponding entry in the same position of the other matrix.

What does the determinant tell me?

The determinant is a single number summarizing a square matrix: it's nonzero exactly when the matrix is invertible, and its absolute value gives the scaling factor the matrix applies to areas or volumes. For a 2×2 matrix [[a,b],[c,d]], it's ad − bc.

What is the inverse of a matrix?

The inverse A⁻¹ undoes A: A·A⁻¹ = I, the identity matrix. Only square matrices with nonzero determinant have inverses. Inverses are central to solving linear systems Ax = b, since x = A⁻¹b.

Matrix Calculator

Perform matrix operations including addition, multiplication, and determinants. Free, fast, accurate — no signup, mobile

Matrix Operations

Operation

Matrix A

Rows

Cols

Matrix B

Rows

Cols

Operation Info

Addition requires matrices of the same dimensions.

Addition Result

[6, 8]
[10, 12]

Trace

Sum of diagonal

Solution Steps

1Matrix Addition: A + B

2Add corresponding elements: (A[i][j] + B[i][j])

How it works

A matrix calculator performs operations on grids of numbers: addition, scalar multiplication, matrix multiplication, transpose, determinant, and inverse. Addition is element-by-element; multiplication combines rows of the first matrix with columns of the second.

Matrix multiplication

(AB)ᵢⱼ = Σₖ Aᵢₖ · Bₖⱼ        (columns of A must equal rows of B)

A, B: the matrices being multiplied
(i, j): row i of A with column j of B

Worked example

A = [[1, 2], [3, 4]]
Multiply by B = [[5, 6], [7, 8]]

Top-left = 1×5 + 2×7 = 19
Continue for each entry

AB = [[19, 22], [43, 50]].

Good to know

Matrix multiplication isn't commutative: AB usually doesn't equal BA.
You can multiply A·B only if A's column count equals B's row count.
The determinant tells you whether a square matrix is invertible (nonzero) or not (zero).