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

Matrix multiplication

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.

Given $n$-vectors $a, b$.

Which of the following are true?

If $ab^T = 0$, then
  1. one of $a$ or $b$ must be $0$
    Correct!
  2. both $a$ and $b$ must be $0$
    Incorrect.
  3. both $a$ and $b$ can be nonzero
    Incorrect.

Suppose $A^T A$ is a diagonal matrix. Then

If $A$ and $B$ are matrices, then $(A+B)^2 =$
  1. $A^2 + 2AB + B^2$
    Incorrect.
  2. $A^2 + 2BA + B^2$
    Incorrect.
  3. $A^2 + AB + BA + B^2$
    Correct!
  4. all of the above
    Incorrect.
  5. none of the above
    Incorrect.

If $A = \left[\begin{array}{ccc}0&0&0 \\ 0&0& -3 \\ 0&0&0 \end{array}\right]$, then
  1. $A^2 = 0$
    Correct!
  2. $A^2 = A$
    Incorrect.
  3. $A^2 = I$
    Incorrect.
  4. $A^2 = -A$
    Incorrect.

If $a$ is real, then $a^2 = -1$ is impossible. Is the matrix analog true or false? For a real-valued matrix $A$,
  1. $A^2 = -I$ is possible
    Correct!
  2. $A^2 = -I$ is not possible
    Incorrect.

A particular computer takes around one second to multiply two square matrices of order 2000. About how long will it take to multiply two square matrices of order 200?
  1. 0.001 second
    Correct!
  2. 0.01 second
    Incorrect.
  3. 0.1 second
    Incorrect.

A matrix whose columns are othonormal is called
  1. an orthonormal matrix
    Incorrect.
  2. a matrix whose columns are othonormal
    Correct!

The product of two lower triangular matrices is
  1. diagonal
    Incorrect.
  2. zero
    Incorrect.
  3. lower triangular
    Correct!
  4. sparse
    Incorrect.

Suppose $A=QR$ is the QR factorization of a square matrix $A$ with independent columns. Which of the following are true?

Suppose the columns of a matrix $A$ are orthonormal, and we (attempt) to compute its QR factorization $A=QR$. Which of the following are true?