Matrix Calculator
Perform matrix operations including addition, multiplication, and determinants. Free, fast, accurate — no signup, mobile
Matrix Operations
Matrix A
Matrix B
Operation Info
Addition Result
[6, 8] [10, 12]
Trace
Sum of diagonal
Solution Steps
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).
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Frequently Asked Questions
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.