Q: Evaluate the 2 by 2 determinant: $$ \begin{vmatrix} 2 & 5 \\ 1 & 4 \end{vmatrix} $$

$$2 \times 4 - 5 \times 1 = 3$$

Q: Find the minor $$M_{13} of $$the following 3 by 3 determinant: $$ \begin{vmatrix} 1 & 3 & 2 \\ 3 & 1 & 4 \\ 3 & 0 & 3 \end{vmatrix} $$

$$(3 \times 0) - (1 \times 3) = -3$$

Question 1

Evaluate the 2 by 2 determinant:

$$
\begin{vmatrix}
2 & 5 \
1 & 4
\end{vmatrix}
$$

Accepted Answer

$$2 	imes 4 - 5 	imes 1 = 3$$

Question 2

Given the system of equations:

$$
\begin{cases}
3x + 2y = 7 \
4x - y = 5
\end{cases}
$$

Use Cramer's Rule to find the value of $$x.$$

Accepted Answer

$$x = 2$$

Question 3

Which of the following is the correct formula for the value of a 2 by 2 determinant $$
\begin{vmatrix}
a & b \
c & d
\end{vmatrix}
$$?

Accepted Answer

$$ad - bc$$

Question 4

A chemist is mixing two solutions. The system of equations below models the number of liters $$x$$ and $$y of $$each solution needed:

$$
\begin{cases}
2x + 3y = 12 \
4x - y = 8
\end{cases}
$$

What is the value of $$y$$ using Cramer's Rule?

Accepted Answer

$$y = 4$$

Question 5

Evaluate the 3 by 3 determinant:

$$
\begin{vmatrix}
2 & 1 & 3 \
0 & 3 & 4 \
5 & 1 & 2
\end{vmatrix}
$$

Accepted Answer

$$29$$

Question 6

For the system of equations:

$$
\begin{cases}
x + 2y = 5 \
2x + 4y = 10
\end{cases}
$$

What can you conclude about the system using determinants?

Accepted Answer

The system is consistent and dependent (infinitely many solutions).

Question 7

If the determinant of the coefficient matrix in a system of three equations is zero, and at least one of the determinants $$D_x$$, $$D_y$$, or $$D_z is $$nonzero, what does this indicate about the system?

Accepted Answer

The system is inconsistent (no solution).

Question 8

Find the minor $$M_{13} of $$the following 3 by 3 determinant:

$$
\begin{vmatrix}
1 & 3 & 2 \
3 & 1 & 4 \
3 & 0 & 3
\end{vmatrix}
$$

Accepted Answer

$$(3 	imes 0) - (1 	imes 3) = -3$$

Question 9

Which property of determinants is illustrated if all entries in a row of a determinant are zero?

Accepted Answer

The determinant is 0.

Question 10

Suppose you swap two rows of a 2 by 2 determinant. What happens to its value?

Accepted Answer

The value changes sign.

Question 11

A medical researcher is analyzing a system of three equations representing concentrations of chemicals in a solution:

$$
\begin{cases}
x + y + z = 6 \
2x - y + 3z = 14 \
-x + 4y - z = -2
\end{cases}
$$

What is the value of the determinant $$D of $$the coefficient matrix?

Accepted Answer

$$D = 12$$

Question 12

If a row of a determinant is multiplied by a nonzero constant $$k$$, what happens to the value of the determinant?

Accepted Answer

It is multiplied by $$k.$$

Question 13

Given the system:

$$
\begin{cases}
2x + 3y = 7 \
4x + 6y = 14
\end{cases}
$$

What is the value of the determinant $$D of $$the coefficient matrix, and what does this imply?

Accepted Answer

$$D = 0$$, so the system is consistent and dependent.

Question 14

Which of the following best describes the cofactor $$A_{ij} of an $$entry $$a_{ij} in a $$determinant?

Accepted Answer

$$A_{ij} = (-1$$)^{i+j}$$ M_{ij}$$, where $$M_{ij} is $$the minor of $$a_{ij}.$$

Question 15

A system of equations has the following coefficient matrix:

$$
\begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
1 & 2 & 3
\end{bmatrix}
$$

What is the value of its determinant?

Accepted Answer

$$0$$

Question 16

If you expand a 3 by 3 determinant across a row with the most zeros, what is the main advantage?

Accepted Answer

It reduces the number of calculations needed.

Question 17

A system of three equations has $$D = 0$$ and $$D_x = D_y = D_z = 0. $$What does this indicate?

Accepted Answer

The system is consistent and dependent (infinitely many solutions).

Question 18

If you add a multiple of one row to another row in a determinant, what happens to the value of the determinant?

Accepted Answer

It remains unchanged.

Question 19

Given the system:

$$
\begin{cases}
2x - y + 3z = 7 \
-x + 4y - z = -2 \
3x + y + 2z = 10
\end{cases}

If $$the determinant $$D of $$the coefficient matrix is nonzero, what does this imply about the system?

Accepted Answer

The system has a unique solution.

Question 20

A business is solving for the prices of three products using the system:

$$
\begin{cases}
2x + y + z = 10 \
x + 3y + 2z = 13 \
3x + 2y + 4z = 20
\end{cases}
$$

What is the value of $$D_x$$ for this system?

Accepted Answer

$$D_x = 6$$

Systems of Linear Equations: Determinants and Cramer's Rule

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