Checkpoint

Determinants, inverses and rank

Twelve questions on determinants, inverses, rank and conditioning, with new numbers and a few classic traps. Work on paper where it helps. Pass at 80% before moving on to eigenvectors, which lean on every idea here.

1
A 2x2 matrix has determinant $-3$ . A triangle of area 4 goes in. What is the area of the triangle that comes out?
area
2
Compute $\det\begin{bmatrix} 2 & 1 & 0 \\ 0 & 3 & 4 \\ 1 & 0 & 5 \end{bmatrix}$ .
det
3
A $3 \times 3$ matrix $\mathbf{A}$ has $\det\mathbf{A} = 2$ . What is $\det(3\mathbf{A})$ ?
4
$\mathbf{A}$ is a $3 \times 3$ matrix with $\det\mathbf{A} = 0$ . Which statement must be true?
5
Find the inverse of $\begin{bmatrix} 4 & 3 \\ 2 & 2 \end{bmatrix}$ .
inverse
6
You apply $\mathbf{B}$ to a vector $\mathbf{x}$ and then $\mathbf{A}$ , which gives $\mathbf{A}\mathbf{B}\mathbf{x}$ . Both matrices are invertible. Which matrix takes $\mathbf{A}\mathbf{B}\mathbf{x}$ back to $\mathbf{x}$ ?
7
Solve $\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}\mathbf{x} = \begin{bmatrix} 4 \\ 11 \end{bmatrix}$ .
x
8
A $3 \times 3$ matrix sends all of 3D space onto a plane through the origin. What are its rank and the dimension of its null space?
9
$\mathbf{B}$ is $1000 \times 4$ and $\mathbf{A}$ is $4 \times 1000$ . What is the largest possible rank of the $1000 \times 1000$ matrix $\mathbf{B}\mathbf{A}$ ?
rank
10
A $2048 \times 5120$ weight matrix gets a LoRA update of rank 8. How many trainable numbers does the update have?
trainable numbers
11
You must solve $\mathbf{A}\mathbf{x} = \mathbf{b}$ where $\mathbf{b}$ comes from slightly noisy measurements. Which matrix is the most dangerous to have as $\mathbf{A}$ ?
12
Code it
Use the determinant's sign as a geometry tool. Write signed_area(p, q, r) for the triangle with corners p, q and r (each a pair of numbers): half the determinant of the 2x2 matrix whose columns are $\mathbf{q} - \mathbf{p}$ and $\mathbf{r} - \mathbf{p}$ , positive when the corners go counterclockwise. Then write orientation(p, q, r), which returns 1 for a counterclockwise turn, -1 for a clockwise turn and 0 for three points on one line (treat a signed area with absolute value at most tol as zero).
⌘↵ runsPython sleeps until you run code

12 still unanswered. Unanswered questions count as wrong.