Chapter 11 Exercises — The Determinant
Work these in order; the tiers build. ⭐ checks the geometric idea, ⭐⭐ is hand computation, ⭐⭐⭐ is proof (math-major / A-track) or coding (C-track), and ⭐⭐⭐⭐ is an application that combines several ideas. Problems marked [proof] ask for a rigorous argument; [code] ask for a numpy-verified implementation. Every numeric answer can be checked with np.linalg.det.
⭐ Tier 1 — Conceptual (the geometric meaning)
11.1. In one sentence each, state what the magnitude, the sign, and the vanishing of $\det(A)$ tell you about the transformation $A$.
11.2. A $2\times 2$ matrix $A$ has $\det(A) = 5$. The unit square is mapped by $A$. What is the area of the image? If instead you feed $A$ a triangle of area $6$, what is the area of the image triangle?
11.3. True or false, with a one-line reason: (a) A rotation has determinant $1$. (b) A matrix with a negative entry has a negative determinant. (c) If $\det(A) = 0$ then $A$ is invertible. (d) Shearing a region changes its area.
11.4. Why does scaling a $2\times 2$ matrix's transformation by a linear factor of $3$ in both directions give determinant $9$ rather than $3$? What would the analogous factor be for a $3\times 3$ uniform scaling by $3$?
11.5. Explain geometrically why a $3\times 3$ matrix whose three columns lie in a common plane must have determinant $0$.
11.6. The matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ has only nonnegative entries, yet its determinant is negative. Resolve the apparent paradox by describing what the transformation does.
⭐⭐ Tier 2 — Hand computation
11.7. Compute by the $2\times 2$ formula: (a) $\det\begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, (b) $\det\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, (c) $\det\begin{bmatrix} 4 & 6 \\ 2 & 3 \end{bmatrix}$. For each, state whether the matrix is invertible and whether it preserves orientation.
11.8. Compute $\det\begin{bmatrix} 2 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 4 \end{bmatrix}$ two ways — by cofactor expansion along the first row, and again along the third column — and confirm you get the same number. (You should get $17$.)
11.9. Compute $\det\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$ in one step using the triangular shortcut. Then change the top-right entry from $3$ to $100$ and recompute — does the determinant change? Explain.
11.10. Use cofactor expansion along the cheapest row or column (the one with the most zeros) to evaluate $\det\begin{bmatrix} 2 & 0 & 0 \\ 5 & 3 & 0 \\ 1 & 4 & -2 \end{bmatrix}$. How many $2\times 2$ minors did you actually have to compute, and why?
11.11. Reduce $\begin{bmatrix} 2 & 4 & 2 \\ 1 & 1 & 0 \\ 3 & 1 & 1 \end{bmatrix}$ to triangular form using only row operations, tracking the effect of each operation on the determinant (note every swap and every row scaling). State the determinant.
11.12. Given $\det(A) = 3$ and $\det(B) = -2$ for $3\times 3$ matrices, evaluate without knowing the matrices: (a) $\det(AB)$, (b) $\det(BA)$, (c) $\det(A^{-1})$, (d) $\det(2A)$ [careful — this is not $2\det(A)$], (e) $\det(A^{\mathsf{T}}B)$.
11.13. Find all values of $k$ for which $\begin{bmatrix} k & 2 \\ 8 & k \end{bmatrix}$ is singular. Interpret each geometrically.
11.14. Compute the area of the triangle with vertices $(1,1)$, $(4,2)$, $(2,5)$ using a single $2\times 2$ determinant of edge vectors. Is the vertex listing counterclockwise or clockwise?
⭐⭐⭐ Tier 3 — Proofs (A-track) and coding (C-track)
11.15. [proof] Complete the geometric derivation of the $2\times 2$ formula sketched in §11.3: draw the parallelogram spanned by $(a,c)$ and $(b,d)$ inside the $(a+b)\times(c+d)$ bounding rectangle, identify the corner pieces, and show that subtracting them leaves area $ad - bc$.
11.16. [proof] Prove that if two rows of a square matrix are identical, then $\det(A) = 0$. (Hint: swap the two equal rows and use the swap rule.)
11.17. [proof] Prove that $\det(A^{-1}) = 1/\det(A)$ for any invertible $A$, using only multiplicativity and $\det(I) = 1$. Then explain in one sentence why this re-proves that a singular matrix has no inverse.
11.18. [proof] Prove the triangular determinant rule: for an upper-triangular $n\times n$ matrix $T$, $\det(T) = t_{11}t_{22}\cdots t_{nn}$. (Hint: induct on $n$ using cofactor expansion down the first column.)
11.19. [proof] Show that for an $n\times n$ matrix and a scalar $c$, $\det(cA) = c^n \det(A)$. Connect your answer to the geometric statement about $n$-dimensional volume.
11.20. [proof] Prove that similar matrices have equal determinants: if $B = P^{-1}AP$ then $\det(B) = \det(A)$. Why does this guarantee that the determinant of a linear transformation does not depend on the chosen basis?
11.21. [code] Implement det_cofactor(A) from scratch (pure Python, recursive Laplace expansion along the first row) as specified in the chapter's Build Your Toolkit callout. Verify against np.linalg.det on five random matrices of sizes $2$ through $5$.
11.22. [code] Implement det_via_lu(A) from scratch (forward elimination with partial pivoting, product of pivots, sign flip per swap). Verify against np.linalg.det, and confirm det_via_lu agrees with your det_cofactor from 11.21.
11.23. [code] Time det_cofactor and det_via_lu on random $n\times n$ matrices for $n = 2, 4, 6, 8, 10$. Plot or tabulate the times and explain, from the $O(n!)$ vs. $O(n^3)$ analysis, why the cofactor version becomes unusable. Estimate (don't run) how long det_cofactor would take on a $15\times 15$ matrix.
⭐⭐⭐⭐ Tier 4 — Applications (combine the ideas)
11.24. [code] Orientation test for a polygon. Write a function is_counterclockwise(points) that takes a list of 2D polygon vertices and returns True if they are listed counterclockwise, using the signed area (a sum of $2\times 2$ determinants of consecutive edge vectors — the "shoelace" formula is a chain of such determinants). Test it on a square listed both ways. Relate the sign to back-face culling in graphics (§11.3 application).
11.25. [code] Change-of-variables factor. The map $(u, v) \mapsto (x, y)$ with $x = 2u + v$, $y = u + 3v$ is linear, with matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}$. (a) Compute $|\det(A)|$ — the factor by which areas (and hence a uniform probability density) are scaled. (b) Confirm numerically: sample $10^6$ points uniformly in the unit square, apply $A$, and check that the area of the image (estimated by its bounding statistics, or by scipy.spatial.ConvexHull) is about $|\det(A)|$. Connect to the Jacobian factor of §11.8.
11.26. [proof + application] Equilibrium uniqueness. A two-good linear market model is $A\mathbf{p} = \mathbf{b}$ with $A = \begin{bmatrix} 4 & -1 \\ -2 & 3 \end{bmatrix}$. (a) Show $\det(A) \neq 0$, so the equilibrium price vector is unique, and solve for $\mathbf{p}$ when $\mathbf{b} = (10, 5)$ — once by Cramer's rule and once by np.linalg.solve, confirming they agree. (b) Find a value of the $(1,1)$ entry that would make the system degenerate (determinant $0$), and explain in economic terms what degeneracy means.
Answers to selected odd-numbered problems
- 11.2. Image of the unit square has area $5$; the triangle of area $6$ maps to area $5 \times 6 = 30$. (Every area scales by $|\det| = 5$.)
- 11.7. (a) $5$, invertible, orientation preserved. (b) $-2$, invertible, orientation reversed. (c) $0$ ($4\cdot3 - 6\cdot2 = 0$), singular, not invertible (the columns are dependent: column 2 is $1.5\times$ column 1).
- 11.9. $\det = 1\cdot4\cdot6 = 24$. Changing the top-right entry to $100$ does not change the determinant — for a triangular matrix only the diagonal matters; the entries above it are irrelevant.
- 11.13. Singular when $k^2 - 16 = 0$, i.e. $k = \pm 4$. At those values the columns become dependent and the transformation collapses the plane onto a line.
- 11.19. $\det(cA) = c^n\det(A)$ because scaling all $n$ rows by $c$ multiplies the determinant by $c$ once per row (row-scaling rule), i.e. $c^n$; geometrically, stretching all $n$ dimensions by $c$ scales $n$-volume by $c^n$.