Free Practice Quiz Question List

8 Applications of Linear Algebra Online Quiz Questions

Use this free practice quiz with 20 questions to review 8 Applications of Linear Algebra, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

A model uses the matrix $A=\begin{bmatrix}1&2\\2&3\end{bmatrix}$ and input vector $\mathbf{x}=\begin{bmatrix}1\\2\end{bmatrix}$ . What is the second entry of $A\mathbf{x}$ ?

A
5
B
8
C
10
D
12

True or false

1 point

Elementary row operations preserve the solution set of a system of linear equations.

A
True
B
False

Written response

1 point

Find the determinant of $\begin{bmatrix}2&1\\3&4\end{bmatrix}$ .

Fill in the blank

1 point

The number of vectors in a basis is the of the vector space.

Choose one

1 point

Which condition guarantees that a square matrix is invertible?

A
It must have all positive entries.
B
Its rows must sum to one.
C
Its determinant must be nonzero.
D
It must be a diagonal matrix.

True or false

1 point

A transformation is linear if it satisfies both additivity and compatibility with scalar multiplication.

A
True
B
False

Choose all

1 point

Select all statements that accurately describe how vectors and matrices are used in applied linear algebra.

A
A vector can represent the state of a system.
B
A matrix can represent a transformation of a vector.
C
A matrix is used only to store unrelated numbers.
D
A vector cannot represent coordinates of a point.

Written response

1 point

What is the term for a nonzero vector whose direction is unchanged when a matrix transformation is applied, although its length or sign may change?

Fill in the blank

1 point

Eigenvalues of a matrix $A$ can be found by solving the characteristic equation .

Choose one

1 point

If $T_1(\mathbf{x})=A_1\mathbf{x}$ is followed by $T_2(\mathbf{x})=A_2\mathbf{x}$ , which matrix represents the composition $T_2\circ T_1$ ?

A
$A_1A_2$
B
$A_2A_1$
C
$A_1+A_2$
D
$A_2-A_1$

Choose all

1 point

Select all statements that correctly describe possible solution sets of a system of linear equations.

A
No solution can indicate inconsistent constraints.
B
Exactly one solution occurs when every variable is determined by a pivot.
C
Infinitely many solutions can occur when the system is consistent and has a free variable.
D
A free variable always makes a system inconsistent.

Choose one

1 point

For the shear matrix $S=\begin{bmatrix}1&k\\0&1\end{bmatrix}$ , which statement is correct?

A
It maps every point to the origin.
B
It changes the area by a factor of $|k|$ .
C
It preserves area while shifting points horizontally according to their height.
D
It changes every horizontal line into a vertical line.

True or false

1 point

Changing the basis changes the underlying linear transformation itself, rather than only changing its matrix representation.

A
True
B
False

Open ended

1 point

In an applied model $A\mathbf{x}=\mathbf{b}$ , explain why a system that is close to singular can be sensitive to measurement or input errors, and state how this should affect interpretation of the result.

Choose one

1 point

A matrix A has dimensions 3 by 2. What dimensions must the vector x have for A x to be defined, and what are the dimensions of the resulting vector?

A
A 3-component vector, producing a 2-component vector
B
A 2-component vector, producing a 3-component vector
C
A 3-component vector, producing a 3-component vector
D
A 2-component vector, producing a 2-component vector

Choose one

1 point

A factory uses the production matrix $\begin{bmatrix}4&1\\2&3\end{bmatrix}$ , whose columns correspond to tables and chairs. If it produces $3$ tables and $2$ chairs, which resource-use vector lists cutting hours first and assembly hours second?

A
The factory uses $10$ cutting hours and $12$ assembly hours.
B
The factory uses $14$ cutting hours and $8$ assembly hours.
C
The factory uses $14$ cutting hours and $12$ assembly hours.
D
The factory uses $12$ cutting hours and $14$ assembly hours.

Written response

1 point

What one-word condition on $\det(A)$ makes a square matrix $A$ invertible?

Choose one

1 point

A laboratory must prepare $10$ liters of a $30\%$ acid solution by mixing a $20\%$ solution with a $50\%$ solution. Which amounts are required?

A
$20\%$ solution: $\frac{10}{3}$ liters; $50\%$ solution: $\frac{20}{3}$ liters
B
$20\%$ solution: $\frac{20}{3}$ liters; $50\%$ solution: $\frac{10}{3}$ liters
C
$20\%$ solution: $5$ liters; $50\%$ solution: $5$ liters
D
$20\%$ solution: $7$ liters; $50\%$ solution: $3$ liters

Choose one

1 point

For the shear transformation

S=\begin{bmatrix}1&k\\0&1\end{bmatrix},

what happens to the area of a region under the transformation?

A
Areas are multiplied by k.
B
Areas are multiplied by $|k|$ .
C
Areas are unchanged.
D
All points are fixed.

Written response

1 point

Suppose $A\mathbf{v}=2\mathbf{v}$ for a nonzero vector $\mathbf{v}$ . What scalar multiplies $\mathbf{v}$ in $A^3\mathbf{v}$ ?