Matrix Determinant & Inverse Calculator (2x2 / 3x3)

For a 2x2 matrix [[a,b],[c,d]], the determinant is ad − bc and the trace is a + d. For a 3x3 matrix, the determinant is found by cofactor expansion along the first row — each entry times the determinant of the 2x2 'minor' left after removing its row and column, with alternating signs — and the trace is the sum of the three diagonal entries.

The inverse exists only when the determinant is nonzero (a 'singular' matrix, determinant zero, has no inverse — this is reported rather than treated as an error, since it's a valid mathematical outcome). For 2x2, the inverse is (1 ÷ det) × [[d,−b],[−c,a]]. For 3x3, it's the adjugate matrix (the transpose of the matrix of cofactors) divided by the determinant — the standard closed-form inverse formula for any invertible square matrix up to 3x3.

When 2x2 is selected, only the top-left four entries (rows/cols 1-2) are used — the remaining fields are ignored, the same way a 'solve for' selector on other calculators on this site ignores whichever field isn't part of the current calculation.