Bring the chapter together: compute determinants of every size, read their geometry, use the row rules and the product rule, and expand along the best line. No worked example sits above the answers.

A mechanic has several ways to test an engine and chooses by the symptom. A determinant can be found by a formula, by elimination or by expansion, and the matrix in front of you says which is quickest.

Decide first which tool fits. A $2\times2$ or $3\times3$ with no pattern: the formula. A line with many zeros: expand along it. Anything larger: row operations to a triangle, keeping track of swaps and scalings. A question about $AB$, $A^{T}$ or $A^{-1}$: the product rule.

• $|\det A|$ is an area or a volume factor, and its sign is orientation.
• Swap: sign changes. Scale a row by $c$: multiply by $c$. Add a multiple of a row: no change.
• $\det A^{T}=\det A$, $\det(cA)=c^n\det A$, and an invertible matrix has $\det A\neq0$.
• Laplace: $\det A=\sum_j a_{ij}C_{ij}$ along any row, with .