Matrix Calculator

Matrix multiplication is an operation that takes two matrices and produces a third matrix. For matrices $A$ (m×n) and $B$ (n×p), the product $AB$ is an m×p matrix where each element $(AB)_{ij}$ is the dot product of row $i$ from $A$ and column $j$ from $B$. Matrix multiplication is not commutative: $AB \neq BA$.

For a 2×2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $\det(A) = ad - bc$. For larger matrices, you can use Laplace expansion (cofactor expansion) or row operations to reduce the matrix to triangular form and then multiply the diagonal elements.

A square matrix has an inverse if and only if its determinant is non-zero ($\det(A) \neq 0$). Such matrices are called invertible or non-singular. The inverse matrix $A^{-1}$ satisfies $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

An eigenvalue $\lambda$ of a square matrix $A$ is a scalar such that there exists a non-zero vector $\vec{v}$ (called an eigenvector) satisfying $A\vec{v} = \lambda\vec{v}$. Eigenvalues are found by solving the characteristic equation $\det(A - \lambda I) = 0$.

Yes, our matrix calculator can handle matrices of various sizes for different operations. Addition and subtraction require matrices of the same dimensions. Multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix.

Yes, our matrix calculator is completely free to use with no limitations. You can perform as many matrix operations as you need.