How to use the Matrix Operations
- Set the dimensions of each matrix.
- Enter the values row by row.
- Choose the operation you need.
- Check the dimension rule — inner dimensions must agree for multiplication.
How the calculation works
Addition and subtraction require identical dimensions and work element by element. Multiplication requires the columns of the first matrix to equal the rows of the second: an m×n times n×p yields m×p, with each entry the dot product of a row and a column. Matrix multiplication is associative but not commutative — AB and BA are usually different matrices and may not both exist.
The determinant is a single scalar measuring how the transformation scales area or volume. A determinant of zero means the matrix is singular: it collapses space onto a lower dimension, has no inverse, and the corresponding system either has no solution or infinitely many. When the determinant is non-zero the inverse exists, and A⁻¹ = adj(A) ÷ det(A), which is how a linear system Ax = b is solved as x = A⁻¹b.
(AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ ; A⁻¹ = adj(A)/det(A), det(A) ≠ 0Source: Standard linear algebra definitions of matrix product, determinant and matrix inverse.
Worked example
Find the determinant and inverse of [[4, 7], [2, 6]].
- det = (4)(6) − (7)(2) = 24 − 14 = 10.
- Non-zero, so the inverse exists.
- Adjugate = [[6, −7], [−2, 4]].
- A⁻¹ = [[0.6, −0.7], [−0.2, 0.4]].
Determinant 10; inverse [[0.6, −0.7], [−0.2, 0.4]], which multiplied by A gives the identity.
Frequently asked questions
Why can't I multiply these matrices?+
The columns of the first must equal the rows of the second. A 2×3 can multiply a 3×4, but not a 2×4.
What does a zero determinant mean?+
The matrix is singular: rows are linearly dependent, there is no inverse, and the system lacks a unique solution.
Is AB the same as BA?+
Almost never. Matrix multiplication is not commutative, and often only one of the two products is even defined.
What is the transpose for?+
Swapping rows and columns; it appears throughout statistics, graphics and least-squares fitting.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.