Math Core

This test covers linear systems and row operations, echelon forms and pivots, vector equations and span, the matrix equation $A\mathbf{x} = \mathbf{b}$ , solution sets in parametric form, and linear independence.

Question 1

Solve the system $2x + y = 7$ , $\;x - y = -1$ . Give your answer as $(x, y)$ .

Enter a point like (2, -3)

Question 2

For what value of $h$ is the system with augmented matrix $\left[\begin{array}{cc|c} 2 & -3 & 5 \\ -4 & h & 1 \end{array}\right]$ inconsistent?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 3

Which matrix is in reduced row echelon form?

Question 4

Row reduce to solve $x_1 + 2x_2 + x_3 = 5$ , $\;2x_1 + 5x_2 + 3x_3 = 13$ , $\;x_1 + 3x_2 + 3x_3 = 12$ . Give $(x_1, x_2, x_3)$ .

Enter a point like (2, -3)

Question 5

How many pivot positions does $\begin{bmatrix} 1 & 2 & 3 & 1 \\ 2 & 4 & 7 & 3 \\ 1 & 2 & 2 & 0 \end{bmatrix}$ have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

Is the system with augmented matrix $\left[\begin{array}{ccc|c} 1 & -1 & 3 & 2 \\ 2 & -1 & 4 & 1 \\ -1 & 3 & -7 & -4 \end{array}\right]$ consistent?

Question 7

The system with augmented matrix $\left[\begin{array}{ccc|c} 2 & 4 & -2 & 6 \\ 1 & 3 & 1 & 5 \\ 3 & 5 & -5 & 7 \end{array}\right]$ has infinitely many solutions. If $x_3 = 2$ , what is $x_1$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

Find weights with $x_1\begin{bmatrix} 2 \\ 1 \end{bmatrix} + x_2\begin{bmatrix} 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \end{bmatrix}$ . Give $(x_1, x_2)$ .

Enter a point like (2, -3)

Question 9

Let $\mathbf{a}_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$ , $\mathbf{a}_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ k \end{bmatrix}$ . For what value of $k$ is $\mathbf{b}$ in $\operatorname{Span}\{\mathbf{a}_1, \mathbf{a}_2\}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

Compute $A\mathbf{x}$ for $A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 3 & -2 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$ . Give the result as an ordered pair.

Enter a point like (2, -3)

Question 11

Let $A = \begin{bmatrix} 1 & 3 \\ -2 & -6 \end{bmatrix}$ . Which statement is true?

Question 12

A homogeneous system $A\mathbf{x} = \mathbf{0}$ has a $3 \times 5$ coefficient matrix with $3$ pivot positions. How many free variables does it have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 13

The solutions of a system $A\mathbf{x} = \mathbf{b}$ are $\mathbf{x} = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + t\begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$ . Find the solution with $x_3 = 2$ , as $(x_1, x_2, x_3)$ .

Enter a point like (2, -3)

Question 14

Are the vectors $\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ , $\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}$ , $\begin{bmatrix} 0 \\ 1 \\ 3 \end{bmatrix}$ linearly independent?

Question 15

For what value of $h$ is $\left\{\begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 4 \\ -1 \\ h + 2 \end{bmatrix}\right\}$ linearly dependent?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.