Matrix Calculator
Add, subtract and multiply matrices up to 5×5, and find the determinant, inverse, transpose, trace and powers.
Matrix A
Matrix B
Matrix Result
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Frequently asked questions
How is a 2x2 matrix determinant calculated?
For a 2x2 matrix [[a, b], [c, d]], the determinant is calculated as (ad - bc).
When can two matrices be multiplied?
Only when the number of columns in the first equals the number of rows in the second: an m x n matrix times an n x p matrix gives an m x p result. Matrix multiplication is also not commutative - AB and BA are generally different, and one may not even be defined when the other is.
What does the determinant tell me?
It is the factor by which the matrix scales area or volume, and its sign tells you whether orientation is flipped. A determinant of zero means the matrix is singular: it collapses space to a lower dimension, has no inverse, and the corresponding system of equations has either no solution or infinitely many.
How is the inverse of a matrix calculated?
For a 2x2 matrix, swap the diagonal entries, negate the off-diagonal ones, and divide by the determinant. For larger matrices, Gauss-Jordan elimination on the matrix augmented with the identity is the practical method; the adjugate formula is mathematically tidy but computationally impractical beyond 3x3. Only square matrices with a non-zero determinant have inverses.