Add, multiply and invert matrices instantly.
Pick a size and an operation, type your numbers into the grids, and see the result matrix, the determinant, and whether your matrix can be inverted, live as you type.
Your matrices
What this means
Operation reference
| Operation | Formula | Requires |
|---|---|---|
| A + B / A − B | elementwise: (A±B)ij = aij ± bij | same size |
| A × B | (AB)ij = Σk aik bkj | cols(A) = rows(B) |
| det(A) 2×2 | ad − bc | square |
| det(A) 3×3 | cofactor / Sarrus expansion | square |
| transpose(A) | (AT)ij = aji | any |
| inverse(A) | adj(A) / det(A) | det(A) ≠ 0 |
| scalar × A | (kA)ij = k · aij | any |
How matrix operations work
What Is a Matrix Determinant?
The determinant of a 2x2 matrix [a, b; c, d] is calculated as ad - bc. It tells you whether the matrix is invertible (determinant is not zero), the scaling factor of the linear transformation, and the signed area of the parallelogram formed by the column vectors. A zero determinant means the matrix is singular and has no inverse.
Matrix Inverse and Transpose
The inverse of a 2x2 matrix is (1/det) * [d, -b; -c, a]. It only exists when the determinant is nonzero. Multiplying a matrix by its inverse yields the identity matrix. The transpose swaps rows and columns: element at position (i,j) moves to (j,i). Both operations are fundamental in solving linear systems of equations.
Eigenvalues of a 2x2 Matrix
Eigenvalues are solutions to the characteristic equation det(A - lambda*I) = 0. For a 2x2 matrix, this yields a quadratic equation: lambda^2 - (a+d)*lambda + (ad-bc) = 0. Eigenvalues describe the directions along which a linear transformation acts by simple scaling, which is essential in physics, engineering, and data science.
Common questions
What does a zero determinant mean?
A zero determinant means the matrix is singular and has no inverse. The rows (or columns) are linearly dependent, meaning one row is a scalar multiple of the other. The transformation collapses space into a lower dimension.
Can this calculator handle 3x3 matrices?
This calculator is designed for 2x2 matrices. For 3x3 or larger matrices, the computation requires cofactor expansion or row reduction, which involves many more elements. Use a specialized linear algebra tool for larger matrices.
What are eigenvalues used for?
Eigenvalues are used in principal component analysis (PCA), vibration analysis, stability analysis of differential equations, Google PageRank, quantum mechanics, and many other fields where understanding the behavior of linear transformations is important.
How do I know if a matrix is invertible?
A matrix is invertible if and only if its determinant is nonzero. If det = 0, the matrix is singular and no inverse exists. Use the determinant operation first to check before computing the inverse.
For learning and quick checks. Displayed values are rounded to 4 decimals, so verify exact figures for high-precision work.