Unit 2 test: Matrix Algebra

Let $A = \begin{bmatrix} 3 & -1 & 2 \\ 0 & 4 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 \\ 2 & -1 \\ -1 & 0 \end{bmatrix}$. What is the $(1, 1)$ entry of $AB$?

$A$ is $2 \times 4$, $B$ is $4 \times 3$ and $C$ is $3 \times 2$. Which product is not defined?

$A$ is $3 \times 2$ and $B$ is $3 \times 4$. Which expression equals $(A^TB)^T$?

Let $A = \begin{bmatrix} 6 & 5 \\ 2 & 2 \end{bmatrix}$. What is the $(1, 2)$ entry of $A^{-1}$?

Use the inverse of the coefficient matrix to solve $\begin{aligned} 7x_1 + 4x_2 &= 10 \\ 5x_1 + 3x_2 &= 7. \end{aligned}$ Enter $(x_1, x_2)$.

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$, the elementary matrix that adds $2$ times row 2 to row 3. What is the $(3, 2)$ entry of $A^{-1}$?

Find the entry in row 2, column 3 of the inverse of $A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$.

For what value of $h$ is $\begin{bmatrix} 1 & -1 & 2 \\ 3 & -2 & h \\ 2 & 1 & 4 \end{bmatrix}$ not invertible?

$A$ is a $6 \times 6$ matrix whose columns are linearly independent. Which statement must be true?

Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 1 & 2 \\ 0 & -2 \\ 3 & -1 \end{bmatrix}$. Find $T(3, 1)$.

$T$ is linear, $T(\mathbf{u}) = (2, -1)$ and $T(\mathbf{v}) = (-1, 2)$. Find $T(3\mathbf{u} - \mathbf{v})$.

Which map $T: \mathbb{R}^2 \to \mathbb{R}^2$ is not linear?

$T: \mathbb{R}^2 \to \mathbb{R}^2$ first reflects points across the $x_2$-axis and then rotates them $90^\circ$ counterclockwise. Find $T(1, 4)$.

Let $T(x_1, x_2, x_3) = (x_1 + 2x_3, \; x_2 - x_3, \; x_1 + x_2 + x_3)$. Which statement is true?