Stewart Precalc 6e Section 10.6: Determinants and Cramer's Rule

1 Concept - Determinant defined for square matrices · Level 1

True or false? $\det(A)$ is defined only for a square matrix $A$.

2 Concept - Determinant is a number · Level 1

True or false? $\det(A)$ is a number, not a matrix.

3 Concept - Singular matrices · Level 1

True or false? If $\det(A) = 0$, then $A$ is not invertible.

4 Concept - Calculating determinants · Level 1

Evaluate the following determinants: (a) $\det(\begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix})$ (b) $\det(\begin{pmatrix} 1 & 0 & 2 \\ 3 & 2 & 1 \\ 0 & -3 & 4 \end{pmatrix})$.

5 Skill - Determinant of 2x2 matrix · Level 1

Find the determinant of the matrix shown in the figure, if it exists.

6 Skill - Determinant of 2x2 matrix · Level 1

Find the determinant of $\begin{pmatrix} 0 & -1 \\ 2 & 0 \end{pmatrix}$, if it exists.

7 Skill - Determinant of 2x2 matrix · Level 1

Find the determinant of $\begin{pmatrix} 4 & 5 \\ 0 & -1 \end{pmatrix}$, if it exists.

8 Skill - Determinant of 2x2 matrix · Level 1

Find the determinant of $\begin{pmatrix} -2 & 1 \\ 3 & -2 \end{pmatrix}$, if it exists.

9 Skill - Determinant of non-square matrix · Level 1

Find the determinant of the column matrix $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$, if it exists.

10 Skill - Determinant of 2x2 matrix · Level 2

Find the determinant of $\begin{pmatrix} \dfrac{1}{2} & \dfrac{1}{8} \\ 1 & \dfrac{1}{2} \end{pmatrix}$, if it exists.

11 Skill - Determinant of 2x2 matrix · Level 2

Find the determinant of $\begin{pmatrix} 2.2 & -1.4 \\ 0.5 & 1.0 \end{pmatrix}$, if it exists.

12 Skill - Minor and cofactor · Level 2

Evaluate the minor $M_{11}$ and cofactor $A_{11}$ using the matrix $A = \begin{pmatrix} 1 & 0 & \dfrac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{pmatrix}$.

13 Skill - Minor and cofactor · Level 2

Evaluate the minor $M_{33}$ and cofactor $A_{33}$ using the matrix $A = \begin{pmatrix} 1 & 0 & \dfrac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{pmatrix}$.

14 Skill - Minor and cofactor · Level 2

Evaluate the minor $M_{12}$ and cofactor $A_{12}$ using the matrix $A = \begin{pmatrix} 1 & 0 & \dfrac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{pmatrix}$.

15 Skill - Minor and cofactor · Level 2

Evaluate the minor $M_{13}$ and cofactor $A_{13}$ using the matrix $A = \begin{pmatrix} 1 & 0 & \dfrac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{pmatrix}$.

16 Skill - Minor and cofactor · Level 2

Evaluate the minor $M_{23}$ and cofactor $A_{23}$ using the matrix $A = \begin{pmatrix} 1 & 0 & \dfrac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{pmatrix}$.

17 Skill - Minor and cofactor · Level 2

Evaluate the minor $M_{32}$ and cofactor $A_{32}$ using the matrix $A = \begin{pmatrix} 1 & 0 & \dfrac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{pmatrix}$.

18 Skill - Determinant of 3x3 matrix · Level 2

Find the determinant of $\begin{pmatrix} 2 & 1 & 0 \\ 0 & -2 & 4 \\ 0 & 1 & -3 \end{pmatrix}$, if it exists.

19 Skill - Determinant of 3x3 matrix · Level 2

Find the determinant of $\begin{pmatrix} 1 & 2 & 5 \\ -2 & -3 & 2 \\ 3 & 5 & 3 \end{pmatrix}$, if it exists.

20 Skill - Determinant of 3x3 matrix · Level 2

Find the determinant of $\begin{pmatrix} 30 & 0 & 20 \\ 0 & -10 & -20 \\ 40 & 0 & 10 \end{pmatrix}$, if it exists.

21 Skill - Determinant of 3x3 matrix · Level 3

Find the determinant of $\begin{pmatrix} -2 & -\dfrac{3}{2} & \dfrac{1}{2} \\ 2 & 4 & 0 \\ \dfrac{1}{2} & 2 & 1 \end{pmatrix}$, if it exists.

22 Skill - Determinant of 3x3 matrix · Level 2

Find the determinant of $\begin{pmatrix} 1 & 3 & 7 \\ 2 & 0 & 8 \\ 0 & 2 & 2 \end{pmatrix}$, if it exists.

23 Skill - Determinant of 3x3 matrix · Level 2

Find the determinant of $\begin{pmatrix} 0 & -1 & 0 \\ 2 & 6 & 4 \\ 1 & 0 & 3 \end{pmatrix}$, if it exists.

24 Skill - Determinant of 4x4 matrix · Level 3

Find the determinant of $\begin{pmatrix} 1 & 3 & 3 & 0 \\ 0 & 2 & 0 & 1 \\ -1 & 0 & 0 & 2 \\ 1 & 6 & 4 & 1 \end{pmatrix}$, if it exists.

25 Skill - Determinant of 4x4 matrix · Level 3

Find the determinant of $\begin{pmatrix} 1 & 2 & 0 & 2 \\ 3 & -4 & 0 & 4 \\ 0 & 1 & 6 & 0 \\ 1 & 0 & 2 & 0 \end{pmatrix}$, if it exists.

26 Skill - Determinant of triangular matrix · Level 2

Evaluate the determinant $\det(\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 2 & 4 & 6 & 8 \\ 0 & 0 & 3 & 6 & 9 \\ 0 & 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & 0 & 5 \end{pmatrix})$ using row or column operations whenever possible.

27 Skill - Cofactor expansion · Level 3

Let $B = \begin{pmatrix} 4 & 1 & 0 \\ -2 & -1 & 1 \\ 4 & 0 & 3 \end{pmatrix}$. (a) Evaluate $\det(B)$ by expanding by the second row. (b) Evaluate $\det(B)$ by expanding by the third column. (c) Do your results in parts (a) and (b) agree?

28 Skill - System with zero determinant · Level 3

Consider the system $\begin{cases} x + 2y + 6z = 5 \\ -3x - 6y + 5z = 8 \\ 2x + 6y + 9z = 7 \end{cases}$. (a) Verify that $x = -1, y = 0, z = 1$ is a solution. (b) Find the determinant of the coefficient matrix. (c) Without solving the system, determine whether there are any other solutions. (d) Can Cramer's Rule be used to solve this system? Why or why not?

29 Skill - Cramer's Rule (2 variables) · Level 2

Use Cramer's Rule to solve the system: $\begin{cases} 2x - y = -9 \\ x + 2y = 8 \end{cases}$.

30 Skill - Cramer's Rule (2 variables) · Level 2

Use Cramer's Rule to solve the system: $\begin{cases} 6x + 12y = 33 \\ 4x + 7y = 20 \end{cases}$.

31 Skill - Cramer's Rule (2 variables) · Level 2

Use Cramer's Rule to solve the system: $\begin{cases} x - 6y = 3 \\ 3x + 2y = 1 \end{cases}$.

32 Skill - Cramer's Rule with fractions · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} \dfrac{1}{2} x + \dfrac{1}{3} y = 1 \\ \dfrac{1}{4} x - \dfrac{1}{6} y = -\dfrac{3}{2} \end{cases}$.

33 Skill - Cramer's Rule (decimal coefficients) · Level 2

Use Cramer's Rule to solve the system: $\begin{cases} 0.4 x + 1.2 y = 0.4 \\ 1.2 x + 1.6 y = 3.2 \end{cases}$.

34 Skill - Cramer's Rule (2 variables) · Level 2

Use Cramer's Rule to solve the system: $\begin{cases} 10 x - 17 y = 21 \\ 20 x - 31 y = 39 \end{cases}$.

35 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} x - y + 2 z = 0 \\ 3 x + z = 11 \\ -x + 2 y = 0 \end{cases}$.

36 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} 5 x - 3 y + z = 6 \\ 4 y - 6 z = 22 \\ 7 x + 10 y = -13 \end{cases}$.

37 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} 2 x_1 + 3 x_2 - 5 x_3 = 1 \\ x_1 + x_2 - x_3 = 2 \\ 2 x_2 + x_3 = 8 \end{cases}$.

38 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} -2 a + c = 2 \\ a + 2 b - c = 9 \\ 3 a + 5 b + 2 c = 22 \end{cases}$.

39 Skill - Cramer's Rule with fractions · Level 4

Use Cramer's Rule to solve the system: $\begin{cases} \dfrac{1}{3} x - \dfrac{1}{5} y + \dfrac{1}{2} z = \dfrac{7}{10} \\ -\dfrac{2}{3} x + \dfrac{2}{5} y + \dfrac{3}{2} z = \dfrac{11}{10} \\ x - \dfrac{4}{5} y + z = \dfrac{9}{5} \end{cases}$.

40 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} 2 x - y = 5 \\ 5 x + 3 z = 19 \\ 4 y + 7 z = 17 \end{cases}$.

41 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} 3 y + 5 z = 4 \\ 2 x - z = 10 \\ 4 x + 7 y = 0 \end{cases}$.

42 Skill - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} 2 x - 5 y = 2 \\ x + y - z = 8 \\ 3 x + 5 z = 0 \end{cases}$.

43 Skill - Cramer's Rule (4 variables) · Level 4

Use Cramer's Rule to solve the system: $\begin{cases} x + y + z + w = 0 \\ 2 x + w = 0 \\ y - z = 0 \\ x + 2 z = 1 \end{cases}$.

44 Skill - Cramer's Rule (4 variables) · Level 4

Use Cramer's Rule to solve the system: $\begin{cases} x + y = 1 \\ y + z = 2 \\ z + w = 3 \\ w - x = 4 \end{cases}$.

45 Skill - Determinant of triangular 5x5 matrix · Level 3

Evaluate the determinant of $\begin{pmatrix} a & a & a & a & a \\ 0 & a & a & a & a \\ 0 & 0 & a & a & a \\ 0 & 0 & 0 & a & a \\ 0 & 0 & 0 & 0 & a \end{pmatrix}$.

46 Skill - Solve for x in determinant equation · Level 3

Solve for $x$: $\det(\begin{pmatrix} x & 12 & 13 \\ 0 & x - 1 & 23 \\ 0 & 0 & x - 2 \end{pmatrix}) = 0$.

47 Skill - Solve for x in determinant equation · Level 3

Solve for $x$: $\det(\begin{pmatrix} x & 1 & 1 \\ 1 & 1 & x \\ x & 1 & x \end{pmatrix}) = 0$.

48 Skill - Solve for x in determinant equation · Level 3

Solve for $x$: $\det(\begin{pmatrix} 1 & 0 & x \\ x^2 & 1 & 0 \\ x & 0 & 1 \end{pmatrix}) = 0$.

49 Skill - Solve for x in determinant equation · Level 3

Solve for $x$: $\det(\begin{pmatrix} a & b & x - a \\ x & x + b & x \\ 0 & 1 & 1 \end{pmatrix}) = 0$.

50 Skill - Triangle area using determinant · Level 2

Sketch the triangle with vertices $(0, 0)$, $(6, 2)$, and $(3, 8)$, and use a determinant to find its area.

51 Skill - Triangle area using determinant · Level 2

Sketch the triangle with vertices $(1, 0)$, $(3, 5)$, and $(-2, 2)$, and use a determinant to find its area.

52 Skill - Triangle area using determinant · Level 2

Sketch the triangle with vertices $(-1, 3)$, $(2, 9)$, and $(5, -6)$, and use a determinant to find its area.

53 Skill - Triangle area using determinant · Level 2

Sketch the triangle with vertices $(-2, 5)$, $(7, 2)$, and $(3, -4)$, and use a determinant to find its area.

54 Skill - Vandermonde determinant · Level 4

Show that $\det(\begin{pmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{pmatrix}) = (x - y)(y - z)(z - x)$.

55 Application - Buying Fruit · Level 3

A roadside fruit stand sells apples at $75 ¢$ a pound, peaches at $90 ¢$ a pound, and pears at $60 ¢$ a pound. Muriel buys 18 pounds of fruit at a total cost of \$13.80. Her peaches and pears together cost \$1.80 more than her apples. (a) Set up a linear system for the number of pounds of apples ($a$), peaches ($p$), and pears ($r$) that she bought. (b) Solve the system using Cramer's Rule.

56 Application - Arch of a Bridge · Level 3

The opening of a railway bridge over a roadway is in the shape of a parabola. A surveyor measures the heights of three points on the bridge as shown in the figure. He wishes to find an equation of the form $y = a x^2 + b x + c$ to model the shape of the arch. (a) Use the surveyed points to set up a system of linear equations for the unknown coefficients $a$, $b$, and $c$. (b) Solve the system using Cramer's Rule.

57 Application - Area of triangular plot · Level 2

An outdoors club is purchasing land to set up a conservation area. The last remaining piece they need to buy is the triangular plot shown in the figure. Use the determinant formula for the area of a triangle to find the area of the plot.

58 Discovery - Proof of triangle area formula · Level 4

The figure shows a triangle in the plane with vertices $(a_1, b_1)$, $(a_2, b_2)$, and $(a_3, b_3)$. (a) Find the coordinates of the vertices of the surrounding rectangle, and find its area. (b) Find the area of the red triangle by subtracting the areas of the three blue triangles from the area of the rectangle. (c) Use your answer to part (b) to show that the area of the red triangle is given by area $= \pm \dfrac{1}{2} \det(\begin{pmatrix} a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \\ a_3 & b_3 & 1 \end{pmatrix})$.

59 Discovery - Collinear points · Level 3

(a) If three points lie on a line, what is the area of the triangle that they determine? Use the answer, together with the determinant formula for the area of a triangle, to explain why the points $(a_1, b_1)$, $(a_2, b_2)$, and $(a_3, b_3)$ are collinear if and only if $\det(\begin{pmatrix} a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \\ a_3 & b_3 & 1 \end{pmatrix}) = 0$. (b) Use a determinant to check whether each given set of points is collinear, and graph them to verify.

60 Discovery - Determinant form of a line · Level 3

(a) Use the result of Exercise 64(a) to show that the equation of the line containing the points $(x_1, y_1)$ and $(x_2, y_2)$ is $\det(\begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix}) = 0$. (b) Use the result of part (a) to find an equation for the line containing the points $(20, 50)$ and $(-10, 25)$.

61 Discovery - Matrices with zero determinant · Level 2

Use the definition of determinant and elementary row/column operations to explain why matrices of the following types have determinant $0$: (a) a matrix with a row or column consisting entirely of zeros; (b) a matrix with two identical rows or two identical columns; (c) a matrix in which one row is a multiple of another row, or one column is a multiple of another column.

62 Discovery - Solving linear systems · Level 2

Suppose you have to solve a linear system with five equations and five variables without the assistance of a calculator or computer. Which method would you prefer: Cramer's Rule or Gaussian elimination? Write a short paragraph explaining the reasons for your answer.

63 Example - Determinant of a 2x2 Matrix · Level 1

Evaluate $|A|$ for $A = \begin{pmatrix} 6 & -3 \\ 2 & 3 \end{pmatrix}$.

64 Example - Determinant of a 3x3 Matrix · Level 2

Evaluate the determinant of the matrix $A = \begin{pmatrix} 2 & 3 & -1 \\ 0 & 2 & 4 \\ -2 & 5 & 6 \end{pmatrix}$.

65 Example - Expanding Determinants by Row and Column · Level 3

Let $A = \begin{pmatrix} 2 & 3 & -1 \\ 0 & 2 & 4 \\ -2 & 5 & 6 \end{pmatrix}$. Evaluate $\det(A)$ by expanding *(a)* by the second row, *(b)* by the third column. Verify that each expansion gives the same value.

66 Example - Invertibility Criterion via Determinant · Level 3

Show that the matrix $A$ has no inverse, where $A = \begin{pmatrix} 1 & 2 & 0 & 4 \\ 0 & 0 & 0 & 3 \\ 5 & 6 & 2 & 6 \\ 2 & 4 & 0 & 9 \end{pmatrix}$.

67 Example - Row and Column Transformations · Level 4

Find the determinant of $A$. Does $A$ have an inverse? $A = \begin{pmatrix} 8 & 2 & -1 & -4 \\ 3 & 5 & -3 & 11 \\ 24 & 6 & 1 & -12 \\ 2 & 2 & 7 & -1 \end{pmatrix}$.

68 Example - Cramer's Rule (Two Variables) · Level 2

Use Cramer's Rule to solve the system: $\begin{cases} 2x + 6y = -1 \\ x + 8y = 2 \end{cases}$.

69 Example - Cramer's Rule (3 variables) · Level 3

Use Cramer's Rule to solve the system: $\begin{cases} 2x - 3y + 4z = 1 \\ x + 6z = 0 \\ 3x - 2y = 5 \end{cases}$.

70 Example - Area of a Triangle · Level 2

Find the area of the triangle shown in Figure 1, which has vertices $(1, 2)$, $(3, 6)$, and $(-1, 4)$.

Quick Answer Input

Submit your exam?