1.4.2—Determinants and inverses

For a 2×2 matrix [[a,b],[c,d]], det A=ad−bc. A square matrix has an inverse only when its determinant is non-zero; determinant magnitude gives the scale factor for oriented area or volume.

A zero determinant means the transformation collapses dimension and distinct vectors become dependent. For larger matrices, expand by a row or column or use row operations while tracking determinant changes.

For [[2,1],[3,2]], det=4−3=1, so the matrix is invertible and preserves area magnitude. For [[1,2],[2,4]], det=0, so no inverse exists.

A negative determinant does not mean “no inverse”; it indicates orientation reversal, while only zero prevents invertibility.