$\newcommand{\ones}{\mathbf 1}$
Matrix multiplication
If $A^TB = 0$ then
every column of $A$ is orthogonal to every column of $B$
Correct!
every row of $A$ is orthogonal to every row of $B$
Incorrect.
Given $n$-vectors $a, b$.
Which of the following are true?
$ab^T = ba^T$
✓
✗
This option is incorrect.
$ab^T = (ba^T)^T$
✓
✗
This option is correct.
$ab^T = b^Ta$
✓
✗
This option is incorrect.
$a^Tb = b^Ta$
✓
✗
This option is correct.
$ab^T = a^Tb$
✓
✗
This option is incorrect.
Submit
If $ab^T = 0$, then
one of $a$ or $b$ must be $0$
Correct!
both $a$ and $b$ must be $0$
Incorrect.
both $a$ and $b$ can be nonzero
Incorrect.
Suppose $A^T A$ is a diagonal matrix. Then
the columns of $A$ are mutually orthogonal
✓
✗
This option is correct.
the columns of $A$ are linearly independent
✓
✗
This option is correct.
the rows of $A$ are mutually orthogonal
✓
✗
This option is incorrect.
the rows of $A$ are linearly independent
✓
✗
This option is incorrect.
Submit
If $A$ and $B$ are matrices, then $(A+B)^2 =$
$A^2 + 2AB + B^2$
Incorrect.
$A^2 + 2BA + B^2$
Incorrect.
$A^2 + AB + BA + B^2$
Correct!
all of the above
Incorrect.
none of the above
Incorrect.
If $A = \left[\begin{array}{ccc}0&0&0 \\ 0&0& -3 \\ 0&0&0 \end{array}\right]$, then
$A^2 = 0$
Correct!
$A^2 = A$
Incorrect.
$A^2 = I$
Incorrect.
$A^2 = -A$
Incorrect.
If $a$ is real, then $a^2 = -1$ is impossible. Is the matrix analog true or false? For a real-valued matrix $A$,
$A^2 = -I$ is possible
Correct!
$A^2 = -I$ 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?
0.001 second
Correct!
0.01 second
Incorrect.
0.1 second
Incorrect.
A matrix whose columns are othonormal is called
an orthonormal matrix
Incorrect.
a matrix whose columns are othonormal
Correct!
The product of two lower triangular matrices is
diagonal
Incorrect.
zero
Incorrect.
lower triangular
Correct!
sparse
Incorrect.
Suppose $A=QR$ is the QR factorization of a square matrix $A$ with independent columns. Which of the following are true?
$Q$ is an orthogonal matrix
✓
✗
This option is correct.
$R$ has positive diagonal entries
✓
✗
This option is correct.
$Q$ has positive diagonal entries
✓
✗
This option is incorrect.
Submit
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?
The QR factorization will fail.
✓
✗
This option is incorrect.
$R=I$
✓
✗
This option is correct.
$R=A$
✓
✗
This option is incorrect.
$Q=I$
✓
✗
This option is incorrect.
$Q=A$
✓
✗
This option is correct.
Submit