Determinants

One Number That Judges a Matrix

The determinant compresses a square matrix into a single number answering: does this matrix destroy information?

Calculations:

• 2×2 case: $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$
• 3×3 case: Rule of Sarrus (three southeast diagonals minus three northeast).

Key properties: Swap rows changes sign; $\det(AB) = \det A \cdot \det B$; $\det A^T = \det A$. A row of zeros or equal rows gives $\det = 0$.

Tip: Check $\det A$ before inverting. If zero, save the work; if tiny, be wary of amplified errors.

The Determinant Is an Area

Geometrically, $|\det A|$ is the area of the parallelogram spanned by the matrix's column vectors. In 3D, it scales volumes. The sign shows orientation (negative = flipped).

$$\text{columns aligned} \iff \text{area} = 0 \iff \det = 0$$

Why singularity destroys information: When $\det = 0$, the transformation squashes the plane onto a line. Two inputs land on one output, so you cannot un-squash it ($A^{-1}$ fails).

Common pitfall: Forgetting absolute value. Area is $|\det A|$; the sign is orientation, not negative area.