Matrix Calculator (exact fractions)

Because the spreadsheet eliminates in binary floating point, where each division rounds and the rounding errors do not quite cancel. A matrix whose rows are dependent has a determinant of exactly zero, and exact rational arithmetic keeps it that way. A tiny nonzero determinant from a float routine usually means singular, but nothing in the number tells you so; here there is no ambiguity.

Yes. Integers, decimals with an optional exponent such as 2.5e-3, and fractions such as -7/12 are all read as exact rationals. A decimal is taken at face value, so 0.1 is exactly one tenth. Results can be shown as fractions or, with the decimals switch, rounded to ten significant digits.

The reduced row echelon form is unique for a given matrix. Its nonzero rows are a clean basis for the row space, their count is the rank, and the columns that hold a leading 1 are the pivot columns, which identify a set of independent columns in the original matrix. It is also how the inverse and the solution of A x = b are computed here.

The tool gives one particular solution, found by setting every free variable to zero, and one null-space vector per free variable. Every solution is the particular one plus some combination of those vectors. When A has full column rank there is no null space and the solution is unique; when rank [A | b] exceeds rank A, there is none.

Exact fractions grow. Elimination on an awkward matrix can produce numerators and denominators with hundreds of digits, and the work grows with them. Twelve by twelve covers coursework, small models and hand-checking a numerical result, while keeping every operation instant. For large numerical problems floating point is the right tool, and this one is the way to check a small case of it.