Matrix calculator
A × B and B × A are generally different matrices, and often one of them does not even exist — a 2×3 can multiply a 3×2 but not the other way round. The inner dimensions have to match, and the result takes the outer ones. That non-commutativity is not a quirk; it is exactly what makes matrices useful for describing rotations, where doing two turns in a different order genuinely lands you somewhere else.
Enter matrices one row per line. Multiplication requires the columns of A to match the rows of B. The determinant exists only for square matrices, and a matrix is invertible only when its determinant is non-zero.
How to use the matrix calculator
The determinant has a geometric meaning that makes it far less abstract: it is the factor by which the matrix scales area in two dimensions, or volume in three. A determinant of 2 doubles areas; a determinant of −1 reflects without changing size; a determinant of zero collapses everything onto a line or a point, which is precisely why such a matrix cannot be inverted. The collapse destroys information that no operation can restore.
Questions
One row per line, values separated by spaces or commas. Every row must have the same length.
When the columns of the first equal the rows of the second. A 2×3 times a 3×2 gives a 2×2.
The factor by which the matrix scales area or volume. Zero means it collapses dimensions.
When its determinant is zero. Such a matrix is called singular.
Almost never. Matrix multiplication is not commutative, and often only one order is even defined.