Compute the determinant of a square matrix via Gaussian elimination with partial pivoting. A non-square, ragged or empty matrix is a clean 400; a singular matrix simply returns 0.

Invert a square matrix via Gauss-Jordan elimination. A non-square, ragged or empty matrix is a clean 400. A singular (non-invertible) matrix is NOT an error — it returns a successful { invertible:false, inverse:null }.

Multiply two matrices A·B. The inner dimensions must agree (columns of A equal rows of B); a mismatch or a ragged/empty matrix is a clean 400.

Solve the linear system A·x = b for a square coefficient matrix A and vector b via Gaussian elimination with partial pivoting. A non-square A or a b of the wrong length is a clean 400. A singular system is NOT an error — it returns a successful { solvable:false, solution:null }.

Transpose a matrix, swapping rows and columns. A ragged/empty matrix is a clean 400.