$\newcommand{\ones}{\mathbf 1}$

Matrices

Let $B = \begin{bmatrix} -1 & 0 & 2 \\ 1 & -2 & 0 \end{bmatrix}$. Which of the following are true?

Suppose $R$ is a matrix. The equation $R=1$
  1. never makes sense
    Incorrect.
  2. is profoundly impolite
    Incorrect.
  3. means all entries of $R$ are $1$
    Incorrect.
  4. implies $R$ has dimensions $1\times 1$
    Correct!
  5. is slang used by people who work in finance
    Incorrect.

Suppose $A=\left[ \begin{array}{cc} 1 & 1 \\ -1 & 0 \end{array}\right]$. What can you say about $A(A(0,-1)+ (2,0))$?
  1. it makes no sense
    Incorrect.
  2. it cannot be determined
    Incorrect.
  3. it is $(1,-1)$
    Correct!
  4. it is $(2,1)$
    Incorrect.
  5. it is $(1,2,1,-1)$
    Incorrect.

Let $s = A^T \textbf{1}$. Then $s_i$, the $i$th element of $s$, is equal to
  1. the sum of the entries in the $i$th row of $A$
    Incorrect.
  2. the sum of the entries in the $i$th column of $A$
    Correct!

Let $P$ be an $m \times n$ matrix and $q$ be an $n$-vector, where $P_{ij}$ is the price of good $j$ in country $i$, and $q_j$ is the quantity of good $j$ needed to produce some product. Then $(Pq)_i$ is
  1. the amount of the product you can afford in country $i$
    Incorrect.
  2. the total cost of the goods needed to produce the product in country $i$
    Correct!

Suppose $A$ is an $n \times n$ matrix. Then $A + A^T$ is symmetric
  1. always
    Correct!
  2. sometimes
    Incorrect.
  3. never
    Incorrect.

Suppose $A$ is an $m\times n$ matrix, and the matrix $Q = \begin{bmatrix} I & A^T \\ A & -I \end{bmatrix}$ makes sense. Then we can conclude
  1. $A$ is tall
    Incorrect.
  2. $A$ is square
    Incorrect.
  3. $A$ is wide
    Incorrect.
  4. none of the above
    Correct!

Let $A$ be a matrix. Then $\begin{bmatrix} A \\ I \end{bmatrix}$ has
  1. linearly independent columns
    Correct!
  2. linearly independent rows
    Incorrect.
  3. both
    Incorrect.
  4. neither
    Incorrect.
  5. none of the above
    Incorrect.

Let $x$ and $y$ be $3$-vectors. For $i=1,2,3$, $y_i$ is the average of $x_1, \ldots, x_i$. Then we have $y=Ax$, with
  1. $A = \left[\begin{array}{ccc}1 & 1 & 1\\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]$
    Incorrect.
  2. $A= \left[\begin{array}{ccc}1 & 0 & 0\\ 1/2 & 1/2 & 0 \\ 1/3 & 1/3 &1/3 \end{array}\right]$
    Correct!
  3. $A = \left[\begin{array}{ccc}1 &1/2 & 1/3\\ 0 & 1 & 1/2 \\ 0 & 0 & 1\end{array}\right]$
    Incorrect.

If $f:\mathbf{R}^n \to \mathbf{R}^m$, with $f(x)=Ax$, then the size of $A$ is
  1. $n \times m$
    Incorrect.
  2. $n \times n$
    Incorrect.
  3. $m \times n$
    Correct!
  4. $m \times m$
    Incorrect.

Suppose $Ax=0$, where $A$ is an $m\times n$ matrix and $x$ is a nonzero $n$-vector. Then
  1. $A=0$
    Incorrect.
  2. the rows of $A$ are linearly dependent
    Incorrect.
  3. the columns of $A$ are linearly dependent
    Correct!

Suppose $A$ is an $m \times n$ matrix for which $Ax=0$ holds for every $n$-vector $x$. Which of the following are true?

If $y = (x, x, x)$ then $y = Ax$ where
  1. $A = \left[\begin{array}{ccc} I & I & I \end{array}\right]$
    Incorrect.
  2. $A = \begin{bmatrix} I \\ I \\ I \end{bmatrix}$
    Correct!
  3. $A = \left[\begin{array}{ccc} I & 0 & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{array}\right]$
    Incorrect.

Row vectors and columns vectors of dimension $n$, with $n>1$,