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||||9𝑏18𝑏7𝑐2𝑐9𝑐6𝑑𝑎7𝑎3𝑏||||
  • B||||9𝑏9𝑏7𝑐7𝑐2𝑐6𝑑6𝑎𝑎3𝑏||||
  • C||||18𝑏9𝑏7𝑐9𝑐7𝑐6𝑑7𝑎6𝑎3𝑏||||
  • D||||9𝑏18𝑏7𝑐7𝑐9𝑐6𝑑6𝑎7𝑎3𝑏||||

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.

  • A315, 45, 90, 270
  • B315, 225, 90, 270
  • C315, 45, 180, 360
  • D315, 225, 180, 360

Q11:

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

  • A43
  • B36
  • C4
  • D4

Q12:

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

  • A5,(𝑎𝑐)(5𝑎5𝑏),(𝑎𝑐)(5𝑎5𝑏)
  • B5,5(𝑎𝑏)(𝑎𝑐),5(𝑎𝑏)(𝑎𝑐)
  • C5,25(𝑎𝑏)(𝑎𝑐),25(𝑎𝑏)(𝑎𝑐)
  • D1,5(𝑎𝑏)(𝑎𝑐),5(𝑎𝑏)(𝑎𝑐)
  • E1,(𝑎𝑐)(5𝑎5𝑏),(𝑎𝑐)(5𝑎5𝑏)

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.

  • A90𝑛=𝑚
  • B18𝑛=𝑚
  • C𝑛=𝑚
  • D15𝑛=𝑚

Q15:

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

  • A2||||14932212532223||||
  • B2||||13249213225232||||
  • C14||||13249213225232||||
  • D7||||13249213225232||||
  • E14||||14932212532223||||

Q16:

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

  • A9(9+18𝑚+7𝑛+2𝑘)
  • B9(9+18𝑚7𝑛2𝑘)
  • C9(9+18𝑚+7𝑛2𝑘)
  • D8118𝑚7𝑛+18𝑘
  • E9(9+18𝑚7𝑛+2𝑘)

Q17:

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

  • A𝑥+8
  • B𝑥
  • C𝑥+6
  • D𝑥+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||||40051407309||||, 5,544
  • B||||45701430009||||, 504
  • C||||45701430009||||, 504
  • D||||400514073099||||, 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(7𝑦3𝑥)(7𝑦+6𝑥)
  • B(7𝑦+6𝑥)(7𝑦3𝑥)
  • C(7𝑦3𝑥)(7𝑦+3𝑥)
  • D(7𝑦+3𝑥)(7𝑦3𝑥)
  • E(7𝑦6𝑥)(7𝑦+3𝑥)

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.

  • A𝑀=2, 𝑀=148, 𝑀𝑀=||||506811525834708245||||
  • B𝑀=54, 𝑀=156, 𝑀𝑀=||||41454184228450||||
  • C𝑀=54, 𝑀=156, 𝑀𝑀=||||506811525834708245||||
  • D𝑀=2, 𝑀=148, 𝑀𝑀=||||41454184228450||||

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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