Calculator tool
How this calculator works
Use the explanation to understand the formula, assumptions, and practical limits behind the calculator result.
What a Matrix Is
A matrix is a rectangular grid of numbers. This calculator performs common matrix operations such as addition, subtraction, multiplication, determinants, and inverses when the entered sizes allow them.
Why Shape Matters
Matrices follow shape rules. Two matrices can be added only when they have the same number of rows and columns. For multiplication, the number of columns in the first matrix must match the number of rows in the second.
A Simple Example
A 2 x 3 matrix can multiply a 3 x 4 matrix because the middle numbers match. The result will be a 2 x 4 matrix. If the middle numbers do not match, multiplication is not defined.
Determinants and Inverses
Only square matrices have determinants. A square matrix has an inverse only when its determinant is not zero. That matters because inverse matrices are used to undo certain transformations and solve some systems of equations.
Frequently asked questions
Why can some matrices not be multiplied?
Because the inner dimensions must match. A 2 x 3 matrix can multiply a 3 x 4 matrix, but not a 2 x 4 matrix.
When does a matrix have an inverse?
A matrix must be square and have a nonzero determinant to be invertible.
Is matrix multiplication commutative?
Usually no. Even when both products exist, and can produce different results.
What does the determinant tell me?
For square matrices, a zero determinant means the matrix is singular and has no inverse.
