Matrix Calculator
Add, multiply and find the determinant of matrices.
Tip: Matrix multiplication is not commutative: A×B ≠ B×A (usually).
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About this digital tool
The matrix calculator performs the basic matrix operations: addition (A + B), subtraction (A − B), multiplication (A × B), the transpose of A and the determinant of A. Choose the operation, fill in the elements of matrix A (and B when needed) and the result appears as a matrix or as a number.
Addition and subtraction are done element by element and require matrices of the same dimensions, while multiplication requires the columns of A to equal the rows of B. The determinant of a 2×2 matrix is ad − bc. For A = [[1, 2], [3, 4]] we have det(A) = 1×4 − 2×3 = −2, and for B = [[5, 6], [7, 8]] the product A × B = [[19, 22], [43, 50]]. It is used in linear algebra, in computer graphics, in economics and in solving systems of equations.
How to use
- Choose the operation: addition, subtraction, multiplication, transpose or determinant.
- Fill in the elements of matrix A.
- For addition, subtraction and multiplication, fill in matrix B too.
- Read the result matrix or the determinant value.
Frequently asked questions
What does the determinant show?
If det ≠ 0, the matrix has an inverse (is invertible). If det = 0, the system may not have a unique solution.
How are two matrices multiplied?
Each element of the result is the sum of the products of a row of A with a column of B. For [[1,2],[3,4]] × [[5,6],[7,8]] the first element is 1×5 + 2×7 = 19.
When can two matrices be added?
Only when they have the same dimensions (the same number of rows and columns). Addition is done element by element.
How is the determinant of a 2×2 matrix calculated?
For [[a, b], [c, d]] the determinant is a×d − b×c. E.g. for [[1, 2], [3, 4]]: 1×4 − 2×3 = −2.