Determinants

What is a Determinant?

The determinant is a single number computed from a square matrix that captures essential geometric and algebraic properties. For a $2 \times 2$ matrix, it is calculated as:

$$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$

Geometrically, the determinant is the signed area scaling factor of the linear transformation. A positive value preserves orientation, a negative value includes a reflection, and zero collapses space entirely.

Core Properties & Pitfalls

The single most important test: a square matrix is invertible if and only if its determinant is nonzero. If $\det A = 0$, the matrix is singular and $A\mathbf{x} = \mathbf{b}$ lacks a unique solution.

Multiplicative property: $$\det(AB) = \det(A)\det(B)$$

Common Pitfalls:

• Calculating the $2 \times 2$ determinant as $ab - cd$ or $ad + bc$ instead of $ad - bc$.
• Believing a zero determinant means invertible. Zero means singular (not invertible).
• Swapping any two rows negates the determinant, while a zero or repeated row makes it 0.