Free Practice Quiz Question List

2 Matrices and Matrix Operations Online Quiz Questions

Use this free practice quiz with 20 questions to review 2 Matrices and Matrix Operations, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

Two matrices can be added only when they have the same dimensions.

A
True
B
False

Written response

1 point

What is the standard term for a matrix that has exactly one row?

Fill in the blank

1 point

If $A$ has size $2\times3$ and $B$ has size $3\times4$ , the product $AB$ has size .

True or false

1 point

For compatible matrices, $(AB)^T=A^TB^T$ .

A
True
B
False

Choose all

1 point

Suppose $A$ and $C$ each have size $2\times3$ , while $B$ has size $3\times4$ . Which products are defined? Select all correct choices.

A
$AB$
B
$BA$
C
$AC$
D
$A^TC$
E
$C^TA$

Fill in the blank

1 point

In Gauss–Jordan elimination for finding an inverse, the augmented matrix is reduced according to $[A\mid I]\longrightarrow[$ $\mid$ $]$ .

Choose all

1 point

Which of the following are valid transpose identities? Select all correct choices.

A
$(A+B)^T=A^T+B^T$
B
$(cA)^T=cA^T$
C
$(AB)^T=A^TB^T$
D
$(A^T)^T=A$

True or false

1 point

With floating-point arithmetic, partial pivoting generally chooses the largest absolute entry in the current pivot column as the pivot.

A
True
B
False

Open ended

1 point

Explain how an LU factorization with pivoting is used to solve $A\mathbf{x}=\mathbf{b}$ , including the two systems that must be solved after the factorization.

Choose one

1 point

Matrix $A$ has size $2\times3$ , and matrix $B$ has size $3\times4$ . What can be concluded about the product $AB$ ?

A
The product is undefined because the outer dimensions differ.
B
The product is defined and has size $2\times4$ .
C
The product is defined and has size $3\times2$ .
D
The product is defined and has size $4\times3$ .

Choose one

1 point

Let $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2&1\\0&3\end{bmatrix}$ . Which matrix equals $AB$ ?

A
$\begin{bmatrix}2&1\\6&15\end{bmatrix}$
B
$\begin{bmatrix}2&7\\15&6\end{bmatrix}$
C
$\begin{bmatrix}2&7\\6&15\end{bmatrix}$
D
$\begin{bmatrix}2&6\\7&15\end{bmatrix}$

Choose one

1 point

Which expression is always equal to $(AB)^T$ , whenever the matrix products are defined?

A
$B^TA^T$
B
$A^TB^T$
C
$BA$
D
$AB$

Choose one

1 point

Consider $A=\begin{bmatrix}2&3\\1&4\end{bmatrix}$ . Which statement correctly determines whether $A$ is invertible?

A
It is singular because its determinant is $0$ .
B
It is nonsquare, so it cannot have an inverse.
C
It is invertible only if all four entries are nonzero.
D
It is invertible because its determinant is $5$ .

Choose one

1 point

The elementary matrix $E=\begin{bmatrix}1&0&0\\3&1&0\\0&0&1\end{bmatrix}$ is multiplied on the left by a $3\times3$ matrix $A$ . What row operation does $EA$ perform on $A$ ?

A
It interchanges rows 1 and 2.
B
It replaces row 2 by $R_2+3R_1$ .
C
It replaces row 1 by $R_1+3R_2$ .
D
It multiplies every row by $3$ .

Written response

1 point

Find the determinant of $A=\begin{bmatrix}3&1\\2&4\end{bmatrix}$ . Enter the value as a whole number.

Written response

1 point

What numerical technique chooses, at each step, a row with the largest absolute value in the current pivot column and interchanges rows before elimination?

Choose one

1 point

To solve $A\mathbf{x}=\mathbf{b}$ by Gaussian elimination, which procedure is appropriate?

A
$[A\mid I]$ , followed by extracting the right block
B
$[I\mid A^{-1}]$ , followed by back-substitution
C
$[A\mid\mathbf{b}]$ , followed by back-substitution after reaching row-echelon form
D
$[A^T\mid\mathbf{b}]$ , followed by taking the transpose

Choose one

1 point

Suppose a system uses an LU factorization with pivoting, $PA=LU$ . Which sequence correctly solves $A\mathbf{x}=\mathbf{b}$ ?

A
First solve $L\mathbf{y}=P\mathbf{b}$ , then solve $U\mathbf{x}=\mathbf{y}$ .
B
First solve $U\mathbf{y}=P\mathbf{b}$ , then solve $L\mathbf{x}=\mathbf{y}$ .
C
First solve $L\mathbf{y}=\mathbf{b}$ , then solve $P U\mathbf{x}=\mathbf{y}$ .
D
First compute $A^{-1}=U^{-1}L^{-1}P^{-1}$ and multiply it by $\mathbf{b}$ .

Choose one

1 point

Let $A=\begin{bmatrix}2&1\\-1&3\end{bmatrix}$ and $\mathbf{x}=\begin{bmatrix}4\\-2\end{bmatrix}$ . What is $A\mathbf{x}$ ?

A
$\begin{bmatrix}10\\-10\end{bmatrix}$
B
$\begin{bmatrix}6\\-10\end{bmatrix}$
C
$\begin{bmatrix}6\\10\end{bmatrix}$
D
$\begin{bmatrix}-6\\-10\end{bmatrix}$

Written response

1 point

Suppose $A$ and $B$ have corresponding entries $a_{21}=-3$ and $b_{21}=12$ , and $cA=B$ . What is the value of the scalar $c$ ?