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

# 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?