Intermediate Algebra 2e · free preview

4.6 Solve Systems of Equations Using Determinants

Learning objectives: Evaluate the determinant of a $2\,\times\,2$ matrix; Evaluate the determinant of a $3\,\times\,3$ matrix; Use Cramer’s Rule to solve systems of equations; Solve applications using determinants

1. Learning objectives

Learning objectives

Learning objectives

  • Evaluate the determinant of a 2 × 22\,\times\,2 matrix
  • Evaluate the determinant of a 3 × 33\,\times\,3 matrix
  • Use Cramer’s Rule to solve systems of equations
  • Solve applications using determinants

In this section we will learn of another method to solve systems of linear equations called Cramer’s rule. Before we can begin to use the rule, we need to learn some new definitions and notation.

2. Evaluate the Determinant of a $2\,\times\,2$ Matrix

Evaluate the Determinant of a 2 × 22\,\times\,2 Matrix

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix [a bc d],[\begin{array}{l} a\,b \\ c\,d \\ \end{array}], we first write it as ∣a bc d∣.|\begin{array}{l} a\,b \\ c\,d \\ \end{array}|. To get the real number value of the determinant we subtract the products of the diagonals, as shown.

The determinant of any square matrix [a bc d],[\begin{array}{l} a\,b \\ c\,d \\ \end{array}], where a, b, c, and d are real numbers, is

∣a bc d∣=ad−bc|\begin{array}{l} a\,b \\ c\,d \\ \end{array}|=ad-bc

3. Example 4.45

Example 4.45

Example 4.45

Evaluate the determinant of ⓐ [4 −23 −1][\begin{array}{l} 4\,-2 \\ 3\,-1 \\ \end{array}] ⓑ [−3−4−20].[\begin{array}{lccr} -3 & & & -4 \\ -2 & & & 0 \\ \end{array}].

Solution

ⓐ

ⓑ

4. Evaluate the Determinant of a $3\,\times\,3$ Matrix

Evaluate the Determinant of a 3 × 33\,\times\,3 Matrix

To evaluate the determinant of a 3 × 33\,\times\,3 matrix, we have to be able to evaluate the minor of an entry in the determinant. The minor of an entry is the 2 × 22\,\times\,2 determinant found by eliminating the row and column in the 3 × 33\,\times\,3 determinant that contains the entry.

The minor of an entry in a 3 × 33\,\times\,3 determinant is the 2 × 22\,\times\,2 determinant found by eliminating the row and column in the 3 × 33\,\times\,3 determinant that contains the entry.

To find the minor of entry a1,{a}_{1}, we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we write the 2 × 22\,\times\,2 determinant that remains.

To find the minor of entry b2,{b}_{2}, we eliminate the row and column that contain it. So we eliminate the 2nd2^{nd} row and 2nd2^{nd} column. Then we write the 2 × 22\,\times\,2 determinant that remains.

5. Example 4.46

Example 4.46

Example 4.46

For the determinant ∣4−2310−3−2−42∣,|\begin{array}{rccrccr} 4 & & & -2 & & & 3 \\ 1 & & & 0 & & & -3 \\ -2 & & & -4 & & & 2 \\ \end{array}|, find and then evaluate the minor of ⓐ a1{a}_{1} ⓑ b3{b}_{3} ⓒ c2.{c}_{2}.

Solution

ⓐ

ⓑ

ⓒ

We are now ready to evaluate a 3 × 33\,\times\,3 determinant. To do this we expand by minors, which allows us to evaluate the 3 × 33\,\times\,3 determinant using 2 × 22\,\times\,2 determinants—which we already know how to evaluate!

To evaluate a 3 × 33\,\times\,3 determinant by expanding by minors along the first row, we use the following pattern:

Remember, to find the minor of an entry we eliminate the row and column that contains the entry.

To evaluate a 3 × 33\,\times\,3 determinant by expanding by minors along the first row, the following pattern:

6. Example 4.47

Example 4.47

Example 4.47

Evaluate the determinant ∣2−3−1320−1−1−2∣|\begin{array}{rccrccr} 2 & & & -3 & & & -1 \\ 3 & & & 2 & & & 0 \\ -1 & & & -1 & & & -2 \\ \end{array}| by expanding by minors along the first row.

Solution

To evaluate a 3 × 33\,\times\,3 determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.

When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.

∣+ − +− + −+ − +∣|\begin{array}{l} +\,-\,+ \\ -\,+\,- \\ +\,-\,+ \\ \end{array}|

When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.

∣+ − +− + −+ − +∣|\begin{array}{l} +\,-\,+ \\ -\,+\,- \\ +\,-\,+ \\ \end{array}|

Notice that the sign pattern in the first row matches the signs between the terms in the expansion by the first row.

Since we can expand by any row or column, how do we decide which row or column to use? Usually we try to pick a row or column that will make our calculation easier. If the determinant contains a 0, using the row or column that contains the 0 will make the calculations easier.

7. Example 4.48

Example 4.48

Example 4.48

Evaluate the determinant ∣4−1−33025−4−3∣|\begin{array}{lccrccr} 4 & & & -1 & & & -3 \\ 3 & & & 0 & & & 2 \\ 5 & & & -4 & & & -3 \\ \end{array}| by expanding by minors.

Solution

To expand by minors, we look for a row or column that will make our calculations easier. Since 0 is in the second row and second column, expanding by either of those is a good choice. Since the second row has fewer negatives than the second column, we will expand by the second row.

8. Use Cramer’s Rule to Solve Systems of Equations

Use Cramer’s Rule to Solve Systems of Equations

Cramer’s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.

Let’s start with the systems of two equations with two variables.

For the system of equations {a1x+b1y=k1a2x+b2y=k2,\mathrm{\{}\begin{array}{l} {a}_{1}x+{b}_{1}y={k}_{1} \\ {a}_{2}x+{b}_{2}y={k}_{2} \\ \end{array}, the solution (x,y)(x,y) can be determined by

Notice that to form the determinant D, we use take the coefficients of the variables.

Notice that to form the determinant Dx{D}_{x} and Dy,{D}_{y}, we substitute the constants for the coefficients of the variable we are finding.

9. Example 4.49

Example 4.49

Example 4.49

How to Solve a System of Equations Using Cramer’s Rule

Solve using Cramer’s Rule: {2x+y=−43x−2y=−6.\mathrm{\{}\begin{array}{l} 2x+y=-4 \\ 3x-2y=-6 \\ \end{array}.

Solve a system of two equations using Cramer’s rule.

  1. Step 1. Evaluate the determinant D, using the coefficients of the variables.
  2. Step 2. Evaluate the determinant Dx.{D}_{x}. Use the constants in place of the x coefficients.
  3. Step 3. Evaluate the determinant Dy.{D}_{y}. Use the constants in place of the y coefficients.
  4. Step 4. Find x and y. x=DxD,x=\frac{{D}_{x}}{D}, y=DyDy=\frac{{D}_{y}}{D}
  5. Step 5. Write the solution as an ordered pair.
  6. Step 6. Check that the ordered pair is a solution to both original equations.

To solve a system of three equations with three variables with Cramer’s Rule, we basically do what we did for a system of two equations. However, we now have to solve for three variables to get the solution. The determinants are also going to be 3 × 33\,\times\,3 which will make our work more interesting!

For the system of equations {a1x+b1y+c1z=k1a2x+b2y+c2z=k2a3x+b3y+c3z=k3,\mathrm{\{}\begin{array}{l} {a}_{1}x+{b}_{1}y+{c}_{1}z={k}_{1} \\ {a}_{2}x+{b}_{2}y+{c}_{2}z={k}_{2} \\ {a}_{3}x+{b}_{3}y+{c}_{3}z={k}_{3} \\ \end{array}, the solution (x,y,z)(x,y,z) can be determined by

10. Example 4.50

Example 4.50

Example 4.50

Solve the system of equations using Cramer’s Rule: {3x−5y+4z=55x+2y+z=02x+3y−2z=3.\mathrm{\{}\begin{array}{l} 3x-5y+4z=5 \\ 5x+2y+z=0 \\ 2x+3y-2z=3 \\ \end{array}.

Solution

Cramer’s rule does not work when the value of the D determinant is 0, as this would mean we would be dividing by 0. But when D=0,D=0, the system is either inconsistent or dependent.

When the value of D=0D=0 and Dx,Dy{D}_{x},{D}_{y} and Dz{D}_{z} are all zero, the system is consistent and dependent and there are infinitely many solutions.

When the value of D=0D=0 and Dx,Dy{D}_{x},{D}_{y} and Dz{D}_{z} are not all zero, the system is inconsistent and there is no solution.

For any system of equations, where the value of the determinant D=0,D=0, ValueofdeterminantsTypeofsystemSolutionD=0 and Dx,Dy and Dz areallzeroconsistentanddependentinfinitelymanysolutionsD=0 and Dx,Dy and Dz arenotallzeroinconsistentnosolution\begin{array}{lcclccl} \mathrm{Value of determinants} & & & \mathrm{Type of system} & & & \mathrm{Solution} \\ D=0\,\mathrm{and}\,{D}_{x},{D}_{y}\,\mathrm{and}\,{D}_{z}\,\mathrm{are all zero} & & & \mathrm{consistent and dependent} & & & \mathrm{infinitely many solutions} \\ D=0\,\mathrm{and}\,{D}_{x},{D}_{y}\,\mathrm{and}\,{D}_{z}\,\mathrm{are not all zero} & & & \mathrm{inconsistent} & & & \mathrm{no solution} \\ \end{array}

In the next example, we will use the values of the determinants to find the solution of the system.

11. Example 4.51

Example 4.51

Example 4.51

Solve the system of equations using Cramer’s rule : {x+3y=4−2x−6y=3.\mathrm{\{}\begin{array}{r} x+3y=4 \\ -2x-6y=3 \\ \end{array}.

Solution

{x+3y=4−2x−6y=3Evaluatethedeterminant D, usingthecoefficientsofthevariables.D=∣13−2−6∣D=−6−(−6)D=0\begin{array}{cccccllccccl} & & & & & \mathrm{\{}\begin{array}{r} x+3y=4 \\ -2x-6y=3 \\ \end{array} \\ \begin{array}{l} \mathrm{Evaluate the determinant}\,D,\,\mathrm{using the} \\ \mathrm{coefficients of the variables.} \\ \end{array} & & & & & D=|\begin{array}{rccr} 1 & & & 3 \\ -2 & & & -6 \\ \end{array}| \\ & & & & & D=-6-(-6) \\ & & & & & D=0 \\ \end{array}

We cannot use Cramer’s Rule to solve this system. But by looking at the value of the determinants Dx{D}_{x} and Dy,{D}_{y}, we can determine whether the system is dependent or inconsistent.

Evaluatethedeterminant Dx. Dx=∣433−6∣ Dx=−24−9 Dx=−33\begin{array}{lcccclcccccl} \mathrm{Evaluate the determinant}\,{D}_{x}. & & & & & \,{D}_{x}=|\begin{array}{lccr} 4 & & & 3 \\ 3 & & & -6 \\ \end{array}| \\ & & & & & \,{D}_{x}=-24-9 \\ & & & & & \,{D}_{x}=-33 \\ \end{array}

Since all the determinants are not zero, the system is inconsistent. There is no solution.

12. Solve Applications using Determinants

Solve Applications using Determinants

An interesting application of determinants allows us to test if points are collinear. Three points (x1,y1),({x}_{1},{y}_{1}),(x2,y2)({x}_{2},{y}_{2}) and (x3,y3)({x}_{3},{y}_{3}) are collinear if and only if the determinant below is zero.

∣x1 y1 1x2 y2 1x3 y3 1∣=0|\begin{array}{l} {x}_{1}\,{y}_{1}\,1 \\ {x}_{2}\,{y}_{2}\,1 \\ {x}_{3}\,{y}_{3}\,1 \\ \end{array}|=0

Three points (x1,y1),({x}_{1},{y}_{1}), (x2,y2)({x}_{2},{y}_{2}) and (x3,y3)({x}_{3},{y}_{3}) are collinear if and only if

∣x1 y1 1x2 y2 1x3 y3 1∣=0|\begin{array}{l} {x}_{1}\,{y}_{1}\,1 \\ {x}_{2}\,{y}_{2}\,1 \\ {x}_{3}\,{y}_{3}\,1 \\ \end{array}|=0

We will use this property in the next example.

13. Example 4.52

Example 4.52

Example 4.52

Determine whether the points (5,−5),(5,-5), (4,−3),(4,-3), and (3,−1)(3,-1) are collinear.

Solution

14. Section 4.6 Exercises

Section 4.6 Exercises

Section 4.6 Exercises

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercises, evaluate the determinant of each square matrix.

Evaluate the Determinant of a 3 × 3 Matrix

In the following exercises, find and then evaluate the indicated minors.

In the following exercises, evaluate each determinant by expanding by minors along the first row.

In the following exercises, evaluate each determinant by expanding by minors.

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

15. Practice 3

  1. Solve using Cramer’s rule: {3x+y=−32x+3y=6.\mathrm{\{}\begin{array}{l} 3x+y=-3 \\ 2x+3y=6 \\ \end{array}.
  1. Solve using Cramer’s rule: {−x+y=22x+y=−4.\mathrm{\{}\begin{array}{l} -x+y=2 \\ 2x+y=-4 \\ \end{array}.
  1. Solve the system of equations using Cramer’s Rule: {3x+8y+2z=−52x+5y−3z=0x+2y−2z=−1.\mathrm{\{}\begin{array}{l} 3x+8y+2z=-5 \\ 2x+5y-3z=0 \\ x+2y-2z=-1 \\ \end{array}.

16. Practice 4

  1. Solve the system of equations using Cramer’s Rule: {3x+y−6z=−32x+6y+3z=03x+2y−3z=−6.\mathrm{\{}\begin{array}{l} 3x+y-6z=-3 \\ 2x+6y+3z=0 \\ 3x+2y-3z=-6 \\ \end{array}.
  1. Solve the system of equations using Cramer’s rule: {4x−3y=88x−6y=14.\mathrm{\{}\begin{array}{l} 4x-3y=8 \\ 8x-6y=14 \\ \end{array}.
  1. Solve the system of equations using Cramer’s rule: {x=−3y+42x+6y=8.\mathrm{\{}\begin{array}{l} x=-3y+4 \\ 2x+6y=8 \\ \end{array}.

17. Practice 5

  1. Determine whether the points (3,−2),(3,-2),(5,−3),(5,-3), and (1,−1)(1,-1) are collinear.
  1. Determine whether the points (−4,−1),(-4,-1),(−6,2),(-6,2), and (−2,−4)(-2,-4) are collinear.
  1. ∣−1−324−2−1−20−3∣|\begin{array}{rccrccr} -1 & & & -3 & & & 2 \\ 4 & & & -2 & & & -1 \\ -2 & & & 0 & & & -3 \\ \end{array}|Find the minor ⓐ a1{a}_{1} ⓑ b1{b}_{1} ⓒ c2{c}_{2}

18. Practice 6

  1. ∣−2−231−30−23−2∣|\begin{array}{rccrccr} -2 & & & -2 & & & 3 \\ 1 & & & -3 & & & 0 \\ -2 & & & 3 & & & -2 \\ \end{array}|Find the minor ⓐ a3{a}_{3} ⓑ b3{b}_{3} ⓒ c3{c}_{3}
  1. ∣4−1−2−3−21−2−57∣|\begin{array}{rccrccr} 4 & & & -1 & & & -2 \\ -3 & & & -2 & & & 1 \\ -2 & & & -5 & & & 7 \\ \end{array}|
  1. ∣4−133−22−104∣|\begin{array}{rccrccr} 4 & & & -1 & & & 3 \\ 3 & & & -2 & & & 2 \\ -1 & & & 0 & & & 4 \\ \end{array}|

19. Practice 7

  1. {x−2y=−52x−3y=−4\mathrm{\{}\begin{array}{l} x-2y=-5 \\ 2x-3y=-4 \\ \end{array}
  1. {2x+y=−43x−2y=−6\mathrm{\{}\begin{array}{l} 2x+y=-4 \\ 3x-2y=-6 \\ \end{array}
  1. {x−3y=−92x+5y=4\mathrm{\{}\begin{array}{l} x-3y=-9 \\ 2x+5y=4 \\ \end{array}

20. Practice 8

  1. {3x+8y=−32x+5y=−3\mathrm{\{}\begin{array}{l} 3x+8y=-3 \\ 2x+5y=-3 \\ \end{array}
  1. {4x−3y+z=72x−5y−4z=33x−2y−2z=−7\mathrm{\{}\begin{array}{l} 4x-3y+z=7 \\ 2x-5y-4z=3 \\ 3x-2y-2z=-7 \\ \end{array}
  1. {11x+9y+2z=−97x+5y+3z=−74x+3y+z=−3\mathrm{\{}\begin{array}{l} 11x+9y+2z=-9 \\ 7x+5y+3z=-7 \\ 4x+3y+z=-3 \\ \end{array}

21. Practice 9

  1. {2x+5y=43y−z=34x+3z=−3\mathrm{\{}\begin{array}{l} 2x+5y=4 \\ 3y-z=3 \\ 4x+3z=-3 \\ \end{array}
  1. {3x−z=−35y+2z=−64x+3y=−8\mathrm{\{}\begin{array}{l} 3x-z=-3 \\ 5y+2z=-6 \\ 4x+3y=-8 \\ \end{array}
  1. {x−4y=−1−3x+12y=3\mathrm{\{}\begin{array}{l} x-4y=-1 \\ -3x+12y=3 \\ \end{array}

22. Practice 10

  1. {4x+3y=220x+15y=5\mathrm{\{}\begin{array}{l} 4x+3y=2 \\ 20x+15y=5 \\ \end{array}
  1. {2x+3y+z=12x+y+z=93x+4y+2z=20\mathrm{\{}\begin{array}{l} 2x+3y+z=12 \\ x+y+z=9 \\ 3x+4y+2z=20 \\ \end{array}
  1. {x−2y+3z=1x+y−3z=73x−4y+5z=7\mathrm{\{}\begin{array}{r} x-2y+3z=1 \\ x+y-3z=7 \\ 3x-4y+5z=7 \\ \end{array}

23. Practice 1

  1. Evaluate the determinant of ⓐ [5 −32 −4][\begin{array}{l} 5\,-3 \\ 2\,-4 \\ \end{array}] ⓑ [−4−607].[\begin{array}{rccr} -4 & & & -6 \\ 0 & & & 7 \\ \end{array}].
  1. Evaluate the determinant of ⓐ [−1 3−2 4][\begin{array}{l} -1\,3 \\ -2\,4 \\ \end{array}] ⓑ [−7−3−50].[\begin{array}{lccr} -7 & & & -3 \\ -5 & & & 0 \\ \end{array}].
  1. For the determinant ∣1−1402−1−2−33∣,|\begin{array}{rccrccr} 1 & & & -1 & & & 4 \\ 0 & & & 2 & & & -1 \\ -2 & & & -3 & & & 3 \\ \end{array}|, find and then evaluate the minor of ⓐ a1{a}_{1} ⓑ b2{b}_{2} ⓒ c3.{c}_{3}.

24. Practice 2

  1. For the determinant ∣−2−1030−1−1−23∣,|\begin{array}{rccrccr} -2 & & & -1 & & & 0 \\ 3 & & & 0 & & & -1 \\ -1 & & & -2 & & & 3 \\ \end{array}|, find and then evaluate the minor of ⓐ a2{a}_{2} ⓑ b3{b}_{3} ⓒ c2.{c}_{2}.
  1. Evaluate the determinant ∣2−1−303−43−4−3∣|\begin{array}{lccrccr} 2 & & & -1 & & & -3 \\ 0 & & & 3 & & & -4 \\ 3 & & & -4 & & & -3 \\ \end{array}| by expanding by minors.
  1. Evaluate the determinant ∣−2−1−3−1224−40∣|\begin{array}{rccrccr} -2 & & & -1 & & & -3 \\ -1 & & & 2 & & & 2 \\ 4 & & & -4 & & & 0 \\ \end{array}| by expanding by minors.

25. Practice 11

  1. (0,−5),(0,-5),(−2,−2),(-2,-2), and (2,−8).(2,-8).
  1. (−2,1),(-2,1),(−4,4),(-4,4), and (0,−2).(0,-2).
  1. Explain what is meant by the minor of an entry in a square matrix.

26. Practice 12

  1. Explain the steps for solving a system of equations using Cramer’s rule.

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