Matrix Calculator - Add, Multiply, Transpose, Determinant, Inverse
Matrix operations
A
B
A grid of numbers that does something
A matrix looks like a table of numbers and behaves like an instruction. Multiply a point by one and the point moves: rotated, stretched, sheared or flattened. That is why matrices turn up wherever many things change together - graphics, statistics, engineering, machine learning - and why the operations here have the shapes they do. Adding combines two instructions; multiplying performs one after the other.
Using it in three steps
- 1 Set the size of each matrix, then fill the cells.
- 2 Choose the operation. Ones that cannot apply to your sizes say so instead of guessing.
- 3 Read the result grid. A determinant returns a single number rather than a matrix.
Why A × B is not B × A
Matrix multiplication is composition: A × B means do B first, then A. Order matters for the same reason it matters in the physical world - rotating a book and then flipping it does not land in the same place as flipping and then rotating. The tool preserves that order faithfully, which is why swapping A and B usually changes the answer and sometimes makes the multiplication impossible altogether.
What the determinant is telling you
The determinant measures how much a matrix scales area or volume, and its sign says whether orientation is flipped. A determinant of 3 triples areas; a determinant of -1 preserves size but mirrors everything. A determinant of zero means the transformation squashes space into something thinner - a plane onto a line, a line onto a point - and information is lost, which is precisely why no inverse exists to bring it back.