Determinants

One Number from a Square Matrix

The determinant is a single number computed from a square matrix that captures essential information about it. For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $$\det = ad - bc.$$ Larger determinants are computed by cofactor expansion along a row or column (or, in practice, via row reduction), but the $2 \times 2$ case shows the essential idea.

Geometrically, the determinant is the signed area (in 2D) or signed volume (in 3D) scaling factor of the linear map the matrix represents. A determinant of 2 doubles areas; of 1 preserves them; of 0 collapses space onto a lower dimension. The sign records orientation: a negative determinant means the transformation includes a reflection (flips orientation).

The single most important fact ties the determinant to invertibility: a square matrix is invertible if and only if its determinant is nonzero. A zero determinant means the matrix squashes space, losing information irreversibly — it is singular (no inverse). Equivalently, $A\mathbf{x} = \mathbf{b}$ has a unique solution exactly when $\det A \neq 0$, and the columns are linearly independent exactly then.

Useful properties: $\det(AB) = \det(A)\det(B)$ (the determinant of a product is the product of determinants); swapping two rows negates the determinant; and a matrix with a repeated or zero row has determinant 0. These make the determinant a compact detector of independence, invertibility, and solvability all at once.

Common pitfall: getting the $2 \times 2$ determinant wrong ($ad - bc$, not $ab - cd$ or $ad + bc$), and reversing the invertibility rule. A matrix is invertible when its determinant is nonzero; a zero determinant means singular (no inverse). Thinking "zero determinant means invertible" is exactly backwards — zero is precisely the non-invertible case.