1. MIT 18.06 Problem Set 7 — how this assessment works

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 7” from MIT OpenCourseWare 18.06 Linear Algebra (Spring 2010) (https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/), licensed under CC BY-NC-SA 4.0. Work through 12 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

2. MIT 18.06 PSET7 Problem 1 — Fibonacci determinants of the 1, 1, -1 tridiagonal matrix (4 pts)

$F_n$ is the determinant of the 1, 1,-1 tridiagonal matrix of order $n$ :

F_2=\begin{vmatrix} 1 &-1 \\ 1 & 1 \end{vmatrix}=2 \qquad F_3=\begin{vmatrix} 1 &-1 & 0 \\ 1 & 1 &-1 \\ 0 & 1 & 1 \end{vmatrix}=3 \qquad F_4=\begin{vmatrix} 1 &-1 & & \\ 1 & 1 &-1 & \\ & 1 & 1 &-1 \\ & & 1 & 1 \end{vmatrix} \neq 4.

Expand in cofactors to show that $F_n=F_{n-1}+F_{n-2}$ . These determinants are Fibonacci numbers 1, 2, 3, 5, 8, 13, ... The sequence usually starts 1, 1, 2, 3 (with two 1's) so our $F_n$ is the usual $F_{n+1}$ .

3. MIT 18.06 PSET7 Problem 2 — Every second Fibonacci number from the 1, 3, 1 recursion (12 pts)

Cofactors of the 1, 3, 1 matrices in Problem 21 give a recursion $S_n=3S_{n-1}-S_{n-2}$ . Amazingly that recursion produces every second Fibonacci number. Here is the challenge.

Show that $S_n$ is the Fibonacci number $F_{2n+2}$ by proving $F_{2n+2}=3F_{2n}-F_{2n-2}$ . Keep using Fibonacci's rule $F_k=F_{k-1}+F_{k-2}$ starting with $k=2n+2$ .

4. MIT 18.06 PSET7 Problem 3 — Pascal matrix with the last entry reduced by 1 (12 pts)

The symmetric Pascal matrices have determinant 1. If I subtract 1 from the $n$ , $n$ entry, why does the determinant become zero? (Use rule 3 or cofactors.)

\det \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & 20 \end{bmatrix}=1 \text{ (known)} \qquad \det \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & \mathbf{19} \end{bmatrix}=\mathbf{0} \text{ (to explain)}.

5. MIT 18.06 PSET7 Problem 4 — Cofactor matrix C and AC^T = (det A)I (4 pts)

Find the cofactors of $A$ and multiply $AC^{T}$ to find $\det A$ :

A=\begin{bmatrix} 1 & 1 & 4 \\ 1 & 2 & 2 \\ 1 & 2 & 5 \end{bmatrix} \qquad \text{and} \qquad C=\begin{bmatrix} 6 &-3 & 0 \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{bmatrix} \qquad \text{and } AC^{T}=\_\_\_.

If you change that 4 to 100, why is $\det A$ unchanged?

6. MIT 18.06 PSET7 Problem 5 — Jacobian of the spherical coordinate transformation (4 pts)

Spherical coordinates $\rho$ , $\phi$ , $\theta$ satisfy $x=\rho \sin \phi \cos \theta$ and $y=\rho \sin \phi \sin \theta$ and $z=\rho \cos \phi$ . Find the 3 by 3 matrix of partial derivatives: $\partial x/\partial \rho$ , $\partial x/\partial \phi$ , $\partial x/\partial \theta$ in row 1. Simplify its determinant to $J=\rho^2 \sin \phi$ . Then $dV$ in spherical coordinates is $\rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$ the volume of an infinitesimal "coordinate box".

7. MIT 18.06 PSET7 Problem 6 — A "Jacobi formula": 2 by 2 times 3 by 3 minors (12 pts)

Suppose $A$ is a 5 by 5 matrix. Its entries in row 1 multiply determinants (cofactors) in rows 2–5 to give the determinant. Can you guess a "Jacobi formula" for $\det A$ using 2 by 2 determinants from rows 1–2 times 3 by 3 determinants from rows 3–5? Test your formula on the-1, 2,-1 tridiagonal matrix that has determinant 6.

8. MIT 18.06 PSET7 Problem 7 — Cauchy–Binet formula for det(AB) (12 pts)

The 2 by 2 matrix $AB=(2 \text{ by } 3)(3 \text{ by } 2)$ has a "Cauchy–Binet formula" for $\det AB$ :

\det AB=\text{sum of (2 by 2 determinants in } A) \ (2 \text{ by 2 determinants in } B).

(a) Guess which 2 by 2 determinants to use from $A$ and $B$ .

(b) Test your formula when the rows of $A$ are 1, 2, 3 and 1, 4, 7 with $B=A^{T}$ .

9. MIT 18.06 PSET7 Problem 8 — What eigenvalues 0, 1, 2 determine (4 pts)

A 3 by 3 matrix $B$ is known to have eigenvalues $0, 1, 2$ . This is information is enough to find three of these (give the answers where possible):

(a) the rank of $B$ ,
(b) the determinant of $B^{T}B$ ,
(c) the eigenvalues of $B^{T}B$ ,
(d) the eigenvalues of $(B^2+I)^{-1}$ .

10. MIT 18.06 PSET7 Problem 9 — Eigenvalues of three 3 by 3 matrices (review) (4 pts)

(Review) Find the eigenvalues of $A$ , $B$ , and $C$ :

A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \end{bmatrix} \quad \text{and} \quad C=\begin{bmatrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{bmatrix}.

11. MIT 18.06 PSET7 Problem 10 — All matrices that diagonalize A and A^{-1} (4 pts)

Describe all matrices $S$ that diagonalize this matrix $A$ (find all eigenvectors):

A=\begin{bmatrix} 4 & 0 \\ 1 & 2 \end{bmatrix},

Then describe all matrices that diagonalize $A^{-1}$ .

12. MIT 18.06 PSET7 Problem 11 — Diagonalizing the transition matrix A_1 (recommended) (4 pts)

(Recommended) Find $\Lambda$ and $S$ to diagonalize $A_1$ in Problem 15:

A_1=\begin{bmatrix} .6 & .9 \\ .4 & .1 \end{bmatrix}.

What is the limit of $\Lambda^k$ as $k\to \infty$ ? What is the limit of $S\Lambda^k S^{-1}$ ? In the columns of the matrix you see the $\underline{\hspace{2cm}}$ .

13. MIT 18.06 PSET7 Problem 12 — Left eigenvectors, the rank-1 sum for A, and a Toeplitz challenge (24 pts)

The transpose of $A=S\Lambda S^{-1}$ is $A^{T}=(S^{-1})^{T}\Lambda S^{T}$ . The eigenvectors in $A^{T}y=\lambda y$ are the columns of that matrix $(S^{-1})^{T}$ . They are often called left eigenvectors. How do you multiply matrices to find this formula for $A$ ?

\text{Sum of rank-1 matrices} \qquad A=S \Lambda S^{-1}=\lambda_1 x_1 y_1^{T}+\cdots+\lambda_n x_n y_n^{T}.

Challenge problem: in MATLAB (and in GNU Octave), the command A=toepliz(v) produces a symmetric matrix in which each descending diagonal (from left to right) is constant and the first row is v. For instance, if $v=[0 \ 1 \ 0 \ 0 \ 0 \ 1]$ , then toepliz(v) is the matrix with 1s on both sides of the main diagonal and on the far corners, and 0s elsewhere. More generally, let $v(n)$ be the vector in $R^n$ with a 1 in the second and last places and 0s elsewhere, and let $A(n)=\text{toepliz}(v(n))$ .

(a) Experiment with $n=5, \ldots, 12$ in MATLAB to see the repeating pattern of $\det A(n)$ .

(b) Expand $\det A(n)$ in terms of cofactors of the first row and in terms of cofactors of the first column. Use the known determinant $C_n$ of problem 5.2.13 to recover the pattern found in part (a).