Worksheet: Properties of Determinants

In this worksheet, we will practice identifying the properties of determinants and using them to simplify problems.

Q1:

Find the value of ||||418636000||||.

Q2:

Using the properties of determinants, find the value of ||||133487035||||.

Q3:

Use the properties of determinants to evaluate ||||1114714573914||||.

Q4:

Without expanding, find the value of the determinant ||||8327182496||||.

Q5:

Using the properties of determinants, find the value of ||||50𝑥𝑥50𝑥𝑥50𝑥𝑥||||.secsectansecsectan

Q6:

Evaluate ||11121415||+||12131516||+||13141617||++||25262829||.

Q7:

Evaluate ||6111||+||5111||+||4111||++||10111||.

Q8:

Which of the following is equal to the determinant ||||9𝑏9𝑏7𝑐2𝑐7𝑐6𝑑𝑎6𝑎3𝑏||||?

  • A | | | | 9 𝑏 1 8 𝑏 7 𝑐 2 𝑐 9 𝑐 6 𝑑 𝑎 7 𝑎 3 𝑏 | | | |
  • B | | | | 9 𝑏 9 𝑏 7 𝑐 7 𝑐 2 𝑐 6 𝑑 6 𝑎 𝑎 3 𝑏 | | | |
  • C | | | | 1 8 𝑏 9 𝑏 7 𝑐 9 𝑐 7 𝑐 6 𝑑 7 𝑎 6 𝑎 3 𝑏 | | | |
  • D | | | | 9 𝑏 1 8 𝑏 7 𝑐 7 𝑐 9 𝑐 6 𝑑 6 𝑎 7 𝑎 3 𝑏 | | | |

Q9:

Consider the equation ||||𝑎𝑏𝑐𝑒𝑓𝑔𝑥𝑦𝑧||||=29.

Evaluate 𝑎|||𝑏𝑐𝑓𝑔|||𝑏||𝑎𝑐𝑒𝑔||+𝑐|||𝑎𝑏𝑒𝑓|||.

Q10:

Consider the equation ||||10052𝑥+202𝑚1+2𝑥||||=0.sincos

Determine all the possible values of 𝑥 given that 0𝑥360.

  • A 3 1 5 , 4 5 , 9 0 , 2 7 0
  • B 3 1 5 , 2 2 5 , 9 0 , 2 7 0
  • C 3 1 5 , 4 5 , 1 8 0 , 3 6 0
  • D 3 1 5 , 2 2 5 , 1 8 0 , 3 6 0

Q11:

Use the properties of determinants to find the value of 𝑘 which makes 𝑥 a factor of the determinant ||||𝑥+3343𝑥1435𝑥+𝑘||||.

  • A 4 3
  • B36
  • C 4
  • D4

Q12:

Consider the equation ||||5𝑎𝑥5𝑏5𝑐5𝑐5𝑏𝑥5𝑎5𝑏5𝑎5𝑐𝑥||||=0. Given that 𝑎+𝑏+𝑐=1, find its solution set.

  • A 5 , ( 𝑎 𝑐 ) ( 5 𝑎 5 𝑏 ) , ( 𝑎 𝑐 ) ( 5 𝑎 5 𝑏 )
  • B 5 , 5 ( 𝑎 𝑏 ) ( 𝑎 𝑐 ) , 5 ( 𝑎 𝑏 ) ( 𝑎 𝑐 )
  • C 5 , 2 5 ( 𝑎 𝑏 ) ( 𝑎 𝑐 ) , 2 5 ( 𝑎 𝑏 ) ( 𝑎 𝑐 )
  • D 1 , 5 ( 𝑎 𝑏 ) ( 𝑎 𝑐 ) , 5 ( 𝑎 𝑏 ) ( 𝑎 𝑐 )
  • E 1 , ( 𝑎 𝑐 ) ( 5 𝑎 5 𝑏 ) , ( 𝑎 𝑐 ) ( 5 𝑎 5 𝑏 )

Q13:

Find, without expanding, the value of the determinant ||||180𝑦𝑦𝑧𝑧180𝑦𝑧𝑧180180𝑦𝑧180180𝑦𝑦𝑧||||.

Q14:

Given that 𝑛=||||68915911724|||| and 𝑚=||||182427905466351020||||, find a relation between 𝑚 and 𝑛 without expanding either determinant.

  • A 9 0 𝑛 = 𝑚
  • B 1 8 𝑛 = 𝑚
  • C 𝑛 = 𝑚
  • D 1 5 𝑛 = 𝑚

Q15:

Select the determinant that is equal to ||||23249423225282114||||.

  • A 2 | | | | 1 4 9 3 2 2 1 2 5 3 2 2 2 3 | | | |
  • B 2 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • C 1 4 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • D 7 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • E 1 4 | | | | 1 4 9 3 2 2 1 2 5 3 2 2 2 3 | | | |

Q16:

Find, in its simplest form, an expression for the determinant ||||92𝑘2𝑚7𝑛2𝑘12𝑚7𝑛2𝑘2𝑚9+7𝑛||||.

  • A 9 ( 9 + 1 8 𝑚 + 7 𝑛 + 2 𝑘 )
  • B 9 ( 9 + 1 8 𝑚 7 𝑛 2 𝑘 )
  • C 9 ( 9 + 1 8 𝑚 + 7 𝑛 2 𝑘 )
  • D 8 1 1 8 𝑚 7 𝑛 + 1 8 𝑘
  • E 9 ( 9 + 1 8 𝑚 7 𝑛 + 2 𝑘 )

Q17:

Select a factor of the determinant||||𝑥48𝑥8𝑥+1𝑥+1548𝑥+8||||.

  • A 𝑥 + 8
  • B 𝑥
  • C 𝑥 + 6
  • D 𝑥 + 9

Q18:

Consider the equation ||||620111418141612||||=298.

Find, without expanding, the value of the determinant ||||216114141811620||||.

Q19:

Put the determinant ||||457841652238|||| in upper triangular form, and find its value.

  • A | | | | 4 0 0 5 1 4 0 7 3 0 9 | | | | , 5 , 5 4 4
  • B | | | | 4 5 7 0 1 4 3 0 0 0 9 | | | | , 504
  • C | | | | 4 5 7 0 1 4 3 0 0 0 9 | | | | , 5 0 4
  • D | | | | 4 0 0 5 1 4 0 7 3 0 9 9 | | | | , 5,544

Q20:

Use the properties of determinants to find ||||6𝑦+4𝑧5𝑥4𝑧5𝑥4𝑧+5𝑥6𝑦4𝑧6𝑦5𝑥+6𝑦04𝑧||||.

Q21:

Consider ||𝑥𝑦𝑧𝑤||=6.

Find the value of |||(𝑥10𝑦)𝑦(𝑧10𝑤)𝑤|||.

Q22:

Find, in its simplest form, an expression for the determinant ||||7𝑦3𝑥3𝑥3𝑥7𝑦3𝑥3𝑥3𝑥7𝑦||||.

  • A ( 7 𝑦 3 𝑥 ) ( 7 𝑦 + 6 𝑥 )
  • B ( 7 𝑦 + 6 𝑥 ) ( 7 𝑦 3 𝑥 )
  • C ( 7 𝑦 3 𝑥 ) ( 7 𝑦 + 3 𝑥 )
  • D ( 7 𝑦 + 3 𝑥 ) ( 7 𝑦 3 𝑥 )
  • E ( 7 𝑦 6 𝑥 ) ( 7 𝑦 + 3 𝑥 )

Q23:

Use the properties of determinants to find the values of 𝑀=||||276372494||||,𝑀=||||229661250||||.

Then, determine which of the following has the value of 𝑀𝑀 but not in triangular form.

  • A 𝑀 = 2 , 𝑀 = 1 4 8 , 𝑀 𝑀 = | | | | 5 0 6 8 1 1 5 2 5 8 3 4 7 0 8 2 4 5 | | | |
  • B 𝑀 = 5 4 , 𝑀 = 1 5 6 , 𝑀 𝑀 = | | | | 4 1 4 5 4 1 8 4 2 2 8 4 5 0 | | | |
  • C 𝑀 = 5 4 , 𝑀 = 1 5 6 , 𝑀 𝑀 = | | | | 5 0 6 8 1 1 5 2 5 8 3 4 7 0 8 2 4 5 | | | |
  • D 𝑀 = 2 , 𝑀 = 1 4 8 , 𝑀 𝑀 = | | | | 4 1 4 5 4 1 8 4 2 2 8 4 5 0 | | | |

Q24:

Use the properties of determinants to evaluate||6262||.

Q25:

Using the properties of determinants, find the value of |||||3𝑥1(𝑥4)(𝑥1)3𝑥+2(𝑥1)(𝑥+2)3𝑥+5(𝑥+2)(𝑥+5)|||||.

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