Every answer is checked, not just computed
An inverse is multiplied back by the original to confirm it gives the identity. A solution to Ax = b is substituted back to confirm it reproduces b. Those checks are shown in the results table, because floating-point elimination on a nearly-singular matrix can produce numbers that look plausible and are not.
Values are also cleaned up: elimination leaves things like 0.9999999999998 and −0 behind, and printing those instead of 1 and 0 makes a correct answer look wrong.
The determinant tells you what to expect
A square matrix with a non-zero determinant is invertible, and Ax = b then has exactly one solution. A determinant of zero means the rows are linearly dependent — the matrix squashes space flat — and there is either no solution or infinitely many.
Which of those two it is depends on b, and the solver says so explicitly rather than returning an error. That distinction is usually the actual point of the exercise.
Row reduction
RREF is the workhorse behind most of this: the inverse comes from reducing [A | I], the rank is the number of pivots, and solving comes from reducing [A | b]. Running it directly is useful for seeing the structure — which columns are pivots, which variables are free.
Frequently asked questions
Why does A × B differ from B × A?
Because matrix multiplication is not commutative — it composes transformations, and doing them in the other order generally gives a different result. For most pairs the two products are not even the same shape.
Is anything uploaded?
No. Your matrices stay on your machine.
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