Question 1

Which of the following statements about the Gaussian elimination method is TRUE?

Accepted Answer

It uses elementary row operations to reduce the augmented matrix to row-echelon form.

Question 2

Given the system:
$$
\begin{cases}
2x - y - z = -2 \
x + 2y + 3z = -1 \
x - 3y - 2z = 3
\end{cases}
$$
What is the solution for $$z$$ after applying Gaussian elimination?

Accepted Answer

$$z = 2$$

Question 3

According to the Kronecker-Capelli theorem, a system $$AX = B is $$inconsistent if:

Accepted Answer

$$r(A) 
eq r([A|B])$$

Question 4

Consider the homogeneous system:
$$
\begin{cases}
x + y + z + t = 0 \
2x + 3y + 4z + 5t = 0 \
4x + 5y + 7z + 8t = 0 \
5x + 7y + 10z + mt = 0
\end{cases}
$$
For which value of $$m$$ does the system have infinitely many solutions?

Accepted Answer

$$m = 12$$

Question 5

Which operation is NOT an elementary row operation in Gaussian elimination?

Accepted Answer

Multiplying a row by zero

Question 6

Given the system:
$$
\begin{cases}
x_1$ - x_2$ + x_3$ = 1 \
2x_1 + x_2$ - x_3$ = 1 \
4x_1 - x_2$ + \alpha x_3$ = 2
\end{cases}
$$
For which value of $$\alpha is $$the system inconsistent?

Accepted Answer

$$\alpha = 1$$

Question 7

A homogeneous system $$AX = 0$$ with $$n$$ unknowns has a nontrivial solution if and only if:

Accepted Answer

$$r(A) \u003c n$$

Question 8

Suppose you are modeling the flow of goods in an economy using a system of linear equations. If the system has more unknowns than equations and $$r(A) = r([A|B]) \u003c n$$, what can you conclude?

Accepted Answer

There are infinitely many solutions.

Question 9

Given the system:
$$
\begin{cases}
x + y + z = m \
x + y - z = n \
4x + 3y = p
\end{cases}
$$
Using Cramer's rule, what is the formula for $$z$$?

Accepted Answer

$$z = \dfrac{m-n}{2}$$

Question 10

Which of the following best describes the solution set of a homogeneous system with $$m = n$$ and $$\det(A) = 0$$?

Accepted Answer

The system has infinitely many solutions.

Question 11

If a system of equations has an augmented matrix $$[A|B]$$ with $$r(A) = r([A|B]) = n$$, what does this imply about the solution?

Accepted Answer

The system has a unique solution.

Question 12

In a real-world scenario, a chemical engineer sets up a system of equations to balance a reaction. If the system is homogeneous and $$r(A) \u003c n$$, what does this mean for the possible solutions?

Accepted Answer

There are infinitely many solutions.

Question 13

Which of the following is NOT a valid step in the Gaussian elimination process?

Accepted Answer

Swapping two columns of the augmented matrix $$[A|B]$$

Question 14

Given the system:
$$
\begin{cases}
x_1$ - x_2$ + x_3$ - x_4$ = 2 \
x_1$ - x_3$ + 2x_4 = 0 \
4x_1 - 2x_2 - x_3$ - x_4$ = 5 \
4x_1 - x_2$ - 3x_3 + 2x_4 = 3
\end{cases}
$$
How many free variables does the general solution have?

Accepted Answer

1 (corresponding to $$x_4)$$

Question 15

If $$\det(A) 
eq 0$$ for a square system $$AX = B$$, what can you say about the solution?

Accepted Answer

There is a unique solution.

Question 16

A medical researcher uses a system of equations to model the interaction of drugs in the bloodstream. If the system is inconsistent, what does this mean in terms of the ranks?

Accepted Answer

$$r(A) 
eq r([A|B])$$

Study Notes: Systems of Linear Equations in College Algebra

Study Guide - Practice Questions

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