Fluid Mechanics Objective Questions Or Mcq

Sample Quiz Title $\newcommand{\ones}{\mathbf 1}$

Matrices

The matrix $\left[ \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right]$ has a left inverse.
  1. True
    Incorrect.
  2. False
    Correct!

If $A^TB = 0$ then
  1. every column of $A$ is orthogonal to every column of $B$.
    Correct!
  2. every row of $A$ is orthogonal to every row of $B$.
    Incorrect.

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

Let $P \in \mathbf{R}^{m \times n}$ and $q \in \mathbf{R}^n$, 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!

Linear Functions
Let $x,y \in \textbf{R}^3$. 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
  1. $A \in \mathbf{R}^{m \times n}$
    Correct!
  2. $A \in \mathbf{R}^{n \times m}$
    Incorrect.

Suppose we have a square matrix $A$ such that each row and each column has exactly one $1$, and the rest of the entries are $0$. What is $A^{100} x$?
  1. A vector whose norm is $100\|x\|$
    Incorrect.
  2. A permutation of $x$
    Correct!
  3. A normalized $x$: $\frac{x}{\|x\|}$
    Incorrect.

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