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4.6 Solve Systems of Equations Using Determinants

Determinants of 2 by 2 and 3 by 3 arrays by expansion, then Cramer's Rule as a formula-driven alternative that also flags dependent systems.

Intermediate Algebra 2e, 4.6 Solve Systems of Equations Using Determinants

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44 exercises

4.6 Exercise 1

Before you get started, take this readiness quiz.

Simplify: 5(−2)−(−4)(1).5 \left(−2\right) - \left(−4\right) \left(1\right) .
If you missed this problem, review the reference.

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4.6 Exercise 2

Simplify: −3(8−10)+(−2)(6−3)−4(−3−(−4)).−3 \left(8 - 10\right) + \left(−2\right) \left(6 - 3\right) - 4 \left(−3 - \left(−4\right)\right) .
If you missed this problem, review the reference.

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4.6 Exercise 3

Simplify: −12−8.\frac{−12}{−8} .
If you missed this problem, review the reference.

2.5

4.6 Exercise 5

Evaluate the determinant of ⓐ [5−32−4]\begin{bmatrix} 5 −3 \\ 2 −4 \end{bmatrix} ⓑ [−4−607].\begin{bmatrix} −4 & & & −6 \\ 0 & & & 7 \end{bmatrix} .

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4.6 Exercise 6

Evaluate the determinant of ⓐ [−13−24]\begin{bmatrix} −1 3 \\ −2 4 \end{bmatrix} ⓑ [−7−3−50].\begin{bmatrix} −7 & & & −3 \\ −5 & & & 0 \end{bmatrix} .

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4.6 Exercise 8

For the determinant |1−1402−1−2−33|,\begin{vmatrix} 1 & & & −1 & & & 4 \\ 0 & & & 2 & & & −1 \\ −2 & & & −3 & & & 3 \end{vmatrix} , find and then evaluate the minor of ⓐ a1a_{1} ⓑ b2b_{2} ⓒ c3.c_{3} .

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4.6 Exercise 9

For the determinant |−2−1030−1−1−23|,\begin{vmatrix} −2 & & & −1 & & & 0 \\ 3 & & & 0 & & & −1 \\ −1 & & & −2 & & & 3 \end{vmatrix} , find and then evaluate the minor of ⓐ a2a_{2} ⓑ b3b_{3} ⓒ c2.c_{2} .

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4.6 Exercise 11

Evaluate the determinant |3−240−1−223−1|,\begin{vmatrix} 3 & & & −2 & & & 4 \\ 0 & & & −1 & & & −2 \\ 2 & & & 3 & & & −1 \end{vmatrix} , by expanding by minors along the first row.

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4.6 Exercise 12

Evaluate the determinant |3−2−22−14−10−3|,\begin{vmatrix} 3 & & & −2 & & & −2 \\ 2 & & & −1 & & & 4 \\ −1 & & & 0 & & & −3 \end{vmatrix} , by expanding by minors along the first row.

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4.6 Exercise 14

Evaluate the determinant |2−1−303−43−4−3|\begin{vmatrix} 2 & & & −1 & & & −3 \\ 0 & & & 3 & & & −4 \\ 3 & & & −4 & & & −3 \end{vmatrix} by expanding by minors.

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4.6 Exercise 15

Evaluate the determinant |−2−1−3−1224−40|\begin{vmatrix} −2 & & & −1 & & & −3 \\ −1 & & & 2 & & & 2 \\ 4 & & & −4 & & & 0 \end{vmatrix} by expanding by minors.

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4.6 Exercise 17

Solve using Cramer’s rule: {3x+y=−32x+3y=6.\begin{cases} 3 x + y = −3 \\ 2 x + 3 y = 6 \end{cases} .

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4.6 Exercise 18

Solve using Cramer’s rule: {−x+y=22x+y=−4.\begin{cases} − x + y = 2 \\ 2 x + y = −4 \end{cases} .

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4.6 Exercise 20

Solve the system of equations using Cramer’s Rule: {3x+8y+2z=−52x+5y−3z=0x+2y−2z=−1.\begin{cases} 3 x + 8 y + 2 z = −5 \\ 2 x + 5 y - 3 z = 0 \\ x + 2 y - 2 z = −1 \end{cases} .

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4.6 Exercise 21

Solve the system of equations using Cramer’s Rule: {3x+y−6z=−32x+6y+3z=03x+2y−3z=−6.\begin{cases} 3 x + y - 6 z = −3 \\ 2 x + 6 y + 3 z = 0 \\ 3 x + 2 y - 3 z = −6 \end{cases} .

2.5

4.6 Exercise 22

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

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4.6 Exercise 23

Solve the system of equations using Cramer’s rule: {4x−3y=88x−6y=14.\begin{cases} 4 x - 3 y = 8 \\ 8 x - 6 y = 14 \end{cases} .

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4.6 Exercise 24

Solve the system of equations using Cramer’s rule: {x=−3y+42x+6y=8.\begin{cases} x = −3 y + 4 \\ 2 x + 6 y = 8 \end{cases} .

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4.6 Exercise 26

Determine whether the points (3,−2),\left(3 , −2\right) , (5,−3),\left(5 , −3\right) , and (1,−1)\left(1 , −1\right) are collinear.

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4.6 Exercise 27

Determine whether the points (−4,−1),\left(−4 , −1\right) , (−6,2),\left(−6 , 2\right) , and (−2,−4)\left(−2 , −4\right) are collinear.

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4.6 Exercise 29

Evaluate the determinant of each square matrix.

[−48−35]\begin{bmatrix} −4 8 \\ −3 5 \end{bmatrix}

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4.6 Exercise 31

Evaluate the determinant of each square matrix.

[−207−5]\begin{bmatrix} −2 & & & 0 \\ 7 & & & −5 \end{bmatrix}

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4.6 Exercise 33

Find and then evaluate the indicated minors.

|−1−324−2−1−20−3|\begin{vmatrix} −1 & & & −3 & & & 2 \\ 4 & & & −2 & & & −1 \\ −2 & & & 0 & & & −3 \end{vmatrix}
Find the minor ⓐ a1a_{1} ⓑ b1b_{1} ⓒ c2c_{2}

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4.6 Exercise 35

Find and then evaluate the indicated minors.

|−2−231−30−23−2|\begin{vmatrix} −2 & & & −2 & & & 3 \\ 1 & & & −3 & & & 0 \\ −2 & & & 3 & & & −2 \end{vmatrix}
Find the minor ⓐ a3a_{3} ⓑ b3b_{3} ⓒ c3c_{3}

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4.6 Exercise 37

Evaluate each determinant by expanding by minors along the first row.

|4−1−2−3−21−2−57|\begin{vmatrix} 4 & & & −1 & & & −2 \\ −3 & & & −2 & & & 1 \\ −2 & & & −5 & & & 7 \end{vmatrix}

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4.6 Exercise 39

Evaluate each determinant by expanding by minors along the first row.

|13−25−640−2−1|\begin{vmatrix} 1 & & & 3 & & & −2 \\ 5 & & & −6 & & & 4 \\ 0 & & & −2 & & & −1 \end{vmatrix}

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4.6 Exercise 41

Evaluate each determinant by expanding by minors.

|4−133−22−104|\begin{vmatrix} 4 & & & −1 & & & 3 \\ 3 & & & −2 & & & 2 \\ −1 & & & 0 & & & 4 \end{vmatrix}

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4.6 Exercise 43

Evaluate each determinant by expanding by minors.

|2−4−35−1−4320|\begin{vmatrix} 2 & & & −4 & & & −3 \\ 5 & & & −1 & & & −4 \\ 3 & & & 2 & & & 0 \end{vmatrix}

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4.6 Exercise 45

Solve each system of equations using Cramer’s Rule.

{x−2y=−52x−3y=−4\begin{cases} x - 2 y = −5 \\ 2 x - 3 y = −4 \end{cases}

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4.6 Exercise 47

Solve each system of equations using Cramer’s Rule.

{2x+y=−43x−2y=−6\begin{cases} 2 x + y = −4 \\ 3 x - 2 y = −6 \end{cases}

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4.6 Exercise 49

Solve each system of equations using Cramer’s Rule.

{x−3y=−92x+5y=4\begin{cases} x - 3 y = −9 \\ 2 x + 5 y = 4 \end{cases}

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4.6 Exercise 51

Solve each system of equations using Cramer’s Rule.

{3x+8y=−32x+5y=−3\begin{cases} 3 x + 8 y = −3 \\ 2 x + 5 y = −3 \end{cases}

2.5

4.6 Exercise 53

Solve each system of equations using Cramer’s Rule.

{4x−3y+z=72x−5y−4z=33x−2y−2z=−7\begin{cases} 4 x - 3 y + z = 7 \\ 2 x - 5 y - 4 z = 3 \\ 3 x - 2 y - 2 z = −7 \end{cases}

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4.6 Exercise 55

Solve each system of equations using Cramer’s Rule.

{11x+9y+2z=−97x+5y+3z=−74x+3y+z=−3\begin{cases} 11 x + 9 y + 2 z = −9 \\ 7 x + 5 y + 3 z = −7 \\ 4 x + 3 y + z = −3 \end{cases}

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4.6 Exercise 57

Solve each system of equations using Cramer’s Rule.

{2x+5y=43y−z=34x+3z=−3\begin{cases} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{cases}

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4.6 Exercise 59

Solve each system of equations using Cramer’s Rule.

{3x−z=−35y+2z=−64x+3y=−8\begin{cases} 3 x - z = −3 \\ 5 y + 2 z = −6 \\ 4 x + 3 y = −8 \end{cases}

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4.6 Exercise 61

Solve each system of equations using Cramer’s Rule.

{x−4y=−1−3x+12y=3\begin{cases} x - 4 y = −1 \\ −3 x + 12 y = 3 \end{cases}

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4.6 Exercise 63

Solve each system of equations using Cramer’s Rule.

{4x+3y=220x+15y=5\begin{cases} 4 x + 3 y = 2 \\ 20 x + 15 y = 5 \end{cases}

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4.6 Exercise 65

Solve each system of equations using Cramer’s Rule.

{2x+3y+z=12x+y+z=93x+4y+2z=20\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}

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4.6 Exercise 67

Solve each system of equations using Cramer’s Rule.

{x−2y+3z=1x+y−3z=73x−4y+5z=7\begin{cases} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{cases}

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4.6 Exercise 69

Determine whether the given points are collinear.

(0,−5),\left(0 , −5\right) , (−2,−2),\left(−2 , −2\right) , and (2,−8).\left(2 , −8\right) .

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4.6 Exercise 71

Determine whether the given points are collinear.

(−2,1),\left(−2 , 1\right) , (−4,4),\left(−4 , 4\right) , and (0,−2).\left(0 , −2\right) .

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4.6 Exercise 73

Explain what is meant by the minor of an entry in a square matrix.

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4.6 Exercise 75

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

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Adapted from Intermediate Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Intermediate Algebra 2e. Imported at revision 77a933c.