Matrix Calculator

Matrix multiplication is row-times-column dot products, so A's columns must equal B's rows: a 2×3 times a 3×4 works (giving 2×4), but a 2×3 times a 2×3 doesn't — there's nothing for the third element of each row to pair with. The result's shape is always (A's rows) × (B's columns). And even when both orders are legal, A×B ≠ B×A in general — matrix multiplication doesn't commute, which is the first genuinely surprising fact of linear algebra.

Two big things. Geometrically, |det| is the scaling factor of the transformation — a determinant of 6 means the matrix stretches areas (2×2) or volumes (3×3) by 6×; a negative sign means orientation flips. Algebraically, det = 0 is the failure flag: the matrix squashes space into a lower dimension, has no inverse, and as a system of equations Ax = b either has no solution or infinitely many — never exactly one.

The calculator computes in decimals (to 4 places) rather than exact fractions — 0.3333 is 1/3, 0.1667 is 1/6. For homework requiring fraction form, recognize the common ones or multiply the whole inverse by the determinant to see the integer cofactor structure. Also worth knowing: an entry shown as 0 that 'should' be something tiny like 1e-15 has been cleaned — that's floating-point residue from elimination, not information.

Write the system as Ax = b (coefficients in A, constants in b); then x = A⁻¹b. Practically: compute A⁻¹ here, then multiply A⁻¹ × b using the multiplication mode with b as an n×1 matrix. If the inverse doesn't exist (det = 0), the system has no unique solution — dependent or contradictory equations. For large systems, elimination beats inversion, but at homework sizes the inverse route is transparent and checkable.