Chapter 11 Quiz — The Determinant
Eleven conceptual checks. Answer each before opening the solution. These test whether you understand the determinant as a volume verdict, not whether you can grind a cofactor expansion.
Q1. What does the determinant of a matrix represent, geometrically, in one sentence?
Answer
It is the signed factor by which the transformation scales area (in 2D) or volume (in 3D, hyper-volume in higher dimensions): the magnitude is how much regions grow or shrink, and the sign records whether space is flipped. *(The whole chapter in one line.)*Q2. A transformation has $\det = 0$. Name the three equivalent things this tells you.
Answer
(1) The transformation **collapses a dimension** — it flattens the input space onto something lower-dimensional (e.g. a plane onto a line). (2) The matrix is **singular**, i.e. **non-invertible**. (3) The columns (equivalently rows) are **linearly dependent**. All three are the same fact: no volume survives, so information is destroyed and the map cannot be undone.Q3. You scale the plane by a factor of $4$ in every direction. Is the determinant $4$? Explain.
Answer
No — it is $16$. You scaled each *length* by $4$, but area is two-dimensional and scales by $4 \times 4 = 16$. For an $n\times n$ uniform scaling by $c$, the determinant is $c^n$, because $n$-volume scales as the $n$-th power of length.Q4. What does a negative determinant signal, and why can you not undo it by rotating the page?
Answer
A negative determinant signals **orientation reversal** — the transformation includes a flip (a reflection), turning a left hand into a right hand or a counterclockwise loop into a clockwise one. You cannot undo a flip by rotating the page flat on a table; you would have to lift the page and turn it over. Rotations keep $\det > 0$; reflections force $\det < 0$.Q5. State the multiplicativity property and explain it geometrically.
Answer
$\det(AB) = \det(A)\det(B)$. Geometrically, $AB$ means "do $B$, then $A$"; $B$ scales every volume by $\det(B)$, and then $A$ scales the already-scaled volume by $\det(A)$, so the combined scaling is the product. Volume scalings compose by multiplication.Q6. Why does the determinant of a triangular matrix equal the product of its diagonal entries? Why don't the off-diagonal entries matter?
Answer
Cofactor-expand down the first column: only the top entry is nonzero, so the determinant factors as (top diagonal entry) $\times$ (determinant of the smaller triangular block), and the recursion peels off the diagonal. The off-diagonal entries belong to *shears*, which preserve volume, so they cannot change the determinant — only the diagonal scaling sets it.Q7. When you reduce a matrix to triangular form to compute its determinant, which row operations change the determinant, and how?
Answer
*Adding a multiple of one row to another* (the main elimination step) leaves the determinant **unchanged** (it's a shear). *Swapping two rows* multiplies it by $-1$ (count every swap). *Scaling a row by $c$* multiplies it by $c$. So $\det(A) = (-1)^{\#\text{swaps}} \times (\text{product of pivots})$ when you use only adds and swaps.Q8. Why is cofactor expansion a poor choice for computing the determinant of a $30\times 30$ matrix? What do you use instead?
Answer
Cofactor expansion costs about $n!$ operations — for $n = 30$ that is astronomically large (far beyond any computer's reach, longer than the age of the universe). Instead, reduce to triangular form via **LU decomposition** and multiply the pivots, which costs only about $n^3/3$ operations. Use cofactors only for tiny or *symbolic* matrices.Q9. Is $\det(A^{\mathsf{T}}) = \det(A)$ always, sometimes, or never? What practical consequence does this have for cofactor expansion?
Answer
**Always** — the transpose has the same determinant as the original. The practical consequence: every statement about rows is also a statement about columns, so you may perform cofactor expansion along **any row or any column** and get the same answer. Choose the row/column with the most zeros.Q10. A reflection matrix and a $90°$ rotation matrix both have $|\det| = 1$. How does the determinant distinguish them?
Answer
By the **sign**. A rotation preserves orientation, so $\det = +1$. A reflection reverses orientation, so $\det = -1$. Both preserve area (magnitude $1$), but the sign tells you whether space was flipped. Magnitude answers "how much does area change?"; sign answers "was it flipped?"Q11. In code, why should you never test a matrix's invertibility with det(A) == 0?