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Section exercises

Verbal

Explain why we can always evaluate the determinant of a square matrix.

A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0.

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Examining Cramer’s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a 2 × 2 matrix.

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Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

The inverse does not exist.

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The determinant of 2 × 2 matrix A is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.

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Algebraic

For the following exercises, find the determinant.

| 2 3 3.1 4 , 000 |

7 , 990.7

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| 1.1 0.6 7.2 0.5 |

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| 1 0 0 0 1 0 0 0 3 |

3

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| 1 4 0 0 2 3 0 0 3 |

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| 1 0 1 0 1 0 1 0 0 |

1

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| 2 3 1 3 4 1 5 6 1 |

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| 2 1 4 4 2 8 2 8 3 |

224

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| 6 1 2 4 3 5 1 9 1 |

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| 5 1 1 2 3 1 3 6 3 |

15

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| 1.1 2 1 4 0 0 4.1 0.4 2.5 |

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| 2 1.6 3.1 1.1 3 8 9.3 0 2 |

17.03

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| 1 2 1 3 1 4 1 5 1 6 1 7 0 0 1 8 |

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For the following exercises, solve the system of linear equations using Cramer’s Rule.

2 x 3 y = −1 4 x + 5 y = 9

( 1 , 1 )

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5 x 4 y = 2 4 x + 7 y = 6

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   6 x 3 y = 2 8 x + 9 y = −1

( 1 2 , 1 3 )

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2 x + 6 y = 12 5 x 2 y = 13

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4 x + 3 y = 23    2 x y = −1

( 2 , 5 )

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10 x 6 y = 2 5 x + 8 y = −1

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4 x 3 y = −3 2 x + 6 y = −4

( 1 , 1 3 )

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4 x 5 y = 7 3 x + 9 y = 0

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4 x + 10 y = 180 3 x 5 y = −105

( 15 , 12 )

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   8 x 2 y = −3 4 x + 6 y = 4

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For the following exercises, solve the system of linear equations using Cramer’s Rule.

     x + 2 y 4 z = 1    7 x + 3 y + 5 z = 26 2 x 6 y + 7 z = 6

( 1 , 3 , 2 )

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5 x + 2 y 4 z = 47      4 x 3 y z = 94     3 x 3 y + 2 z = 94

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     4 x + 5 y z = −7 −2 x 9 y + 2 z = 8            5 y + 7 z = 21

( 1 , 0 , 3 )

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4 x 3 y + 4 z = 10 5 x 2 z = 2 3 x + 2 y 5 z = 9

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4 x 2 y + 3 z = 6       6 x + y = 2 2 x + 7 y + 8 z = 24

( 1 2 , 1 , 2 )

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5 x + 2 y z = 1 7 x 8 y + 3 z = 1.5 6 x 12 y + z = 7

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   13 x 17 y + 16 z = 73 11 x + 15 y + 17 z = 61    46 x + 10 y 30 z = 18

( 2 , 1 , 4 )

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4 x 3 y 8 z = 7    2 x 9 y + 5 z = 0.5    5 x 6 y 5 z = 2

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   4 x 6 y + 8 z = 10 2 x + 3 y 4 z = 5           x + y + z = 1

Infinite solutions

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4 x 6 y + 8 z = 10 2 x + 3 y 4 z = 5 12 x + 18 y 24 z = 30

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Technology

For the following exercises, use the determinant function on a graphing utility.

| 1 0 8 9 0 2 1 0 1 0 3 0 0 2 4 3 |

24

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| 1 0 2 1 0 −9 1 3 3 0 −2 −1 0 1 1 −2 |

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| 1 2 1 7 4 0 1 2 100 5 0 0 2 2,000 0 0 0 2 |

1

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| 1 0 0 0 2 3 0 0 4 5 6 0 7 8 9 0 |

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Real-world applications

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

Two numbers add up to 56. One number is 20 less than the other.

Yes; 18, 38

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Two numbers add up to 104. If you add two times the first number plus two times the second number, your total is 208

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Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.

Yes; 33, 36, 37

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Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.

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For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had $10,710 in your combined accounts. How much was invested in each account?

$7,000 in first account, $3,000 in second account.

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Questions & Answers

how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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