Unit 3 test: Determinants

Compute $\begin{vmatrix} -4 & 3 \\ 2 & 5 \end{vmatrix}$.

Compute $\det \begin{bmatrix} 3 & 0 & 1 \\ 2 & -1 & 4 \\ 5 & 2 & 0 \end{bmatrix}$.

Let $A = \begin{bmatrix} 4 & -1 & 3 \\ 2 & 6 & -5 \\ 7 & 0 & 1 \end{bmatrix}$. Find the cofactor $C_{12}$.

Compute $\det \begin{bmatrix} 0 & 2 & 0 & 0 \\ 1 & 5 & 3 & -2 \\ 0 & 4 & 0 & 3 \\ 2 & -1 & 1 & 6 \end{bmatrix}$.

Which matrix has determinant $0$?

Let $A$ be a $3 \times 3$ matrix with $\det A = 6$. The matrix $B$ is obtained from $A$ by these steps, in order: add $4$ times row 1 to row 2; swap rows 1 and 3; multiply row 2 by $-\dfrac{1}{2}$. Find $\det B$.

Use row operations to compute $\det \begin{bmatrix} 1 & 2 & -1 & 3 \\ 2 & 5 & 0 & 4 \\ -1 & -1 & 3 & 2 \\ 3 & 6 & -2 & 11 \end{bmatrix}$.

Let $A$ and $B$ be $3 \times 3$ matrices with $\det A = 9$ and $\det B = -2$. Find $\det(3A^{-1}B^T)$.

Find all values of $h$ for which $\begin{bmatrix} 1 & 2 & 3 \\ 2 & h & 5 \\ 1 & 2 & h \end{bmatrix}$ is not invertible.

Let $A$ be a $3 \times 3$ matrix with $\det A = 0$. Which statement must be true?

Use Cramer's rule to solve $\begin{aligned} 5x_1 + 3x_2 &= 4 \\ 2x_1 - x_2 &= -5 \end{aligned}$.

Use Cramer's rule to solve

Find the area of the parallelogram with vertices $(-1, 0)$, $(4, 1)$, $(1, 4)$ and $(6, 5)$.

Find the volume of the parallelepiped spanned by $(1, 2, 0)$, $(0, 3, 1)$ and $(2, 0, 5)$.

Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix}$, and let $S$ be a square with side length $3$. Find the area of $T(S)$.