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

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

Q: For the system: $$ \begin{cases} 2x - 4y = 8 \\ 5x + y = 13 \end{cases} $$ What is the value of the determinant $$D of $$the coefficients?

$$D = 2 \times 1 - 5 \times (-4) = 2 + 20 = 22$$

Q: Evaluate the 3 by 3 determinant: $$ \begin{vmatrix} 2 & 1 & 3 \\ 0 & 3 & 4 \\ 5 & 1 & 2 \end{vmatrix} $$

$$2(3 \times 2 - 4 \times 1) - 1(0 \times 2 - 4 \times 5) + 3(0 \times 1 - 3 \times 5) = 2(6 - 4) - 1(0 - 20) + 3(0 - 15) = 2(2) - 1(-20) + 3(-15) = 4 + 20 - 45 = -21$$

Q: Given the system: $$ \begin{cases} x + 2y - 3z = 1 \\ 2x - y + z = 15 \\ 3x - y - 2z = -3 \end{cases} $$ What is the value of the determinant $$D of $$the coefficients?

$$D = \begin{vmatrix} 1 & 2 & -3 \\ 2 & -1 & 1 \\ 3 & -1 & -2 \end{vmatrix} = 1((-1)(-2) - 1(-1)) - 2(2(-2) - 1 \times 3) + (-3)(2(-1) - (-1) \times 3) = 1(2 + 1) - 2(-4 - 3) + (-3)(-2 + 3) = 3 + 14 - 3 = 14$$

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

$$M_{13} = \begin{vmatrix} 3 & 1 \\ 3 & 0 \end{vmatrix} = 3 \times 0 - 1 \times 3 = -3$$

Q: Suppose you have the determinant: $$ \begin{vmatrix} 5 & 2 \\ 1 & 7 \end{vmatrix} If $$you swap the two rows, what is the value of the new determinant?

$$-(5 \times 7 - 2 \times 1) = -33$$

Q: A chemist is analyzing a reaction with three unknowns, $$x$$, $$y$$, and $$z$$, and sets up the following system: $$ \begin{cases} 2x + y + 3z = 10 \\ x - y + 2z = 5 \\ 3x + 2y + z = 12 \end{cases} $$ Using Cramer's Rule, what is the value of $$D_x$$?

$$D_x = \begin{vmatrix} 10 & 1 & 3 \\ 5 & -1 & 2 \\ 12 & 2 & 1 \end{vmatrix}$$

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

For the system:
$$
\begin{cases}
2x - 4y = 8 \ 5x + y = 13
\end{cases}
$$
What is the value of the determinant $$D of $$the coefficients?

Accepted Answer

$$D = 2 	imes 1 - 5 	imes (-4) = 2 + 20 = 22$$

Question 4

Which of the following statements about Cramer's Rule is TRUE?

Accepted Answer

Cramer's Rule can only be used when the determinant of the coefficient matrix is nonzero.

Question 5

Evaluate the 3 by 3 determinant:
$$
\begin{vmatrix} 2 & 1 & 3 \ 0 & 3 & 4 \ 5 & 1 & 2 \end{vmatrix}
$$

Accepted Answer

$$2(3 	imes 2 - 4 	imes 1) - 1(0 	imes 2 - 4 	imes 5) + 3(0 	imes 1 - 3 	imes 5) = 2(6 - 4) - 1(0 - 20) + 3(0 - 15) = 2(2) - 1(-20) + 3(-15) = 4 + 20 - 45 = -21$$

Question 6

Given the system:
$$
\begin{cases}
x + 2y - 3z = 1 \ 2x - y + z = 15 \ 3x - y - 2z = -3
\end{cases}
$$
What is the value of the determinant $$D of $$the coefficients?

Accepted Answer

$$D = \begin{vmatrix} 1 & 2 & -3 \ 2 & -1 & 1 \ 3 & -1 & -2 \end{vmatrix} = 1((-1)(-2) - 1(-1)) - 2(2(-2) - 1 	imes 3) + (-3)(2(-1) - (-1) 	imes 3) = 1(2 + 1) - 2(-4 - 3) + (-3)(-2 + 3) = 3 + 14 - 3 = 14$$

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 can you conclude about the system?

Accepted Answer

The system is inconsistent and has no solution.

Question 8

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

Accepted Answer

$$M_{13} = \begin{vmatrix} 3 & 1 \ 3 & 0 \end{vmatrix} = 3 	imes 0 - 1 	imes 3 = -3$$

Question 9

Which property of determinants is illustrated by the following scenario: If two rows of a determinant are identical, the value of the determinant is zero.

Accepted Answer

Theorem (14): If two rows are identical, the determinant is zero.

Question 10

Suppose you have the determinant:
$$
\begin{vmatrix} 5 & 2 \ 1 & 7 \end{vmatrix}

If $$you swap the two rows, what is the value of the new determinant?

Accepted Answer

$$-(5 	imes 7 - 2 	imes 1) = -33$$

Question 11

A chemist is analyzing a reaction with three unknowns, $$x$$, $$y$$, and $$z$$, and sets up the following system:
$$
\begin{cases}
2x + y + 3z = 10 \ x - y + 2z = 5 \ 3x + 2y + z = 12
\end{cases}
$$
Using Cramer's Rule, what is the value of $$D_x$$?

Accepted Answer

$$D_x = \begin{vmatrix} 10 & 1 & 3 \ 5 & -1 & 2 \ 12 & 2 & 1 \end{vmatrix}$$

Question 12

If all entries in the second row of a 3 by 3 determinant are zero, what is the value of the determinant?

Accepted Answer

The determinant is zero.

Question 13

A medical researcher is modeling the interaction of three drugs with the system:
$$
\begin{cases}
x + y + z = 6 \ 2x - y + 3z = 14 \ x + 4y - z = 2
\end{cases}

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

Accepted Answer

The system has a unique solution.

Question 14

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

Accepted Answer

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

Question 15

If you multiply the first row of a 2 by 2 determinant by a nonzero constant $$k$$, how does the value of the determinant change?

Accepted Answer

The determinant is multiplied by $$k.$$

Question 16

A system of equations has $$D = 0$$ and $$D_x = D_y = D_z = 0. $$What can you conclude?

Accepted Answer

The system is consistent and dependent, with infinitely many solutions.

Question 17

Which of the following is a correct application of Theorem (16) regarding determinants?

Accepted Answer

Multiplying a row by a constant and adding it to another row does not change the value of the determinant.

Question 18

In a real-world scenario, an engineer is solving a system of three equations using Cramer's Rule. She finds that $$D = 0$$ and $$D_x = 5. $$What does this indicate about the system?

Accepted Answer

The system is inconsistent and has no solution.

Systems of Linear Equations: Determinants and Cramer's Rule

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