Matrix Calculator

How Matrix Operations Are Calculated

Addition and subtraction combine two matrices of identical dimensions entry by entry: each result cell is simply the sum or difference of the corresponding cells in A and B. Multiplication is different — for A x B, the entry in row i, column j of the result is the dot product of row i of A and column j of B, which is only defined when the number of columns in A equals the number of rows in B. The determinant is computed here by cofactor expansion along the first row (for 2x2, it's the classic ad - bc). The inverse uses the adjugate method: A^-1 = (1 / det(A)) x adj(A), where adj(A) is the transpose of the cofactor matrix — this only exists when det(A) is nonzero, which is why a singular matrix has no inverse.

A Common Mix-Up: Matrix Multiplication Is Not Commutative

Unlike ordinary number multiplication, A x B usually does not equal B x A for matrices — the order matters, and swapping it can even change whether the multiplication is defined at all. If you're working through a system of linear equations or a transformation pipeline, always keep track of which matrix is applied first. If your underlying task is really about solving equations rather than matrix algebra itself, the quadratic formula calculator may be a faster route for a single unknown.

When the Determinant Is Zero

A determinant of zero means the matrix is "singular" — it has no inverse, and geometrically it means the transformation collapses space into a lower dimension (a 3x3 matrix with a zero determinant flattens 3D space onto a plane or line). If you need a single scalar count or ratio result rather than full matrix output, the scientific calculator handles the underlying arithmetic for one-off computations.