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 ||||41−8−636000||||.

Q2:

Using the properties of determinants, find the value of ||||−1334−8−703−5||||.

Q3:

Use the properties of determinants to evaluate ||||11147−14−57−3914||||.

Q4:

Without expanding, find the value of the determinant ||||8−3−271−824−9−6||||.

Q5:

Using the properties of determinants, find the value of ||||50𝑥𝑥50𝑥𝑥50𝑥−𝑥||||.secsectansecsectan

Q6:

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

Q7:

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

Q8:

Which of the following is equal to the determinant ||||−9𝑏9𝑏7𝑐2𝑐−7𝑐−6𝑑−𝑎6𝑎−3𝑏||||?

  • A||||−9𝑏−18𝑏7𝑐2𝑐9𝑐−6𝑑−𝑎−7𝑎−3𝑏||||
  • B||||9𝑏−9𝑏7𝑐−7𝑐2𝑐−6𝑑6𝑎−𝑎−3𝑏||||
  • C||||−18𝑏9𝑏7𝑐9𝑐−7𝑐−6𝑑−7𝑎6𝑎−3𝑏||||
  • D||||9𝑏−18𝑏7𝑐−7𝑐9𝑐−6𝑑6𝑎−7𝑎−3𝑏||||

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 0≤𝑥≤360∘∘.

  • 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+3−34−3𝑥−1−43−5𝑥+𝑘||||.

  • A43
  • B36
  • C−4
  • D4

Q12:

Consider the equation ||||5𝑎−𝑥5𝑏5𝑐5𝑐5𝑏−𝑥5𝑎5𝑏5𝑎5𝑐−𝑥||||=0. Given that 𝑎+𝑏+𝑐=−1, find its solution set.

  • A−5,√(𝑎−𝑐)(5𝑎−5𝑏),−√(𝑎−𝑐)(5𝑎−5𝑏)
  • B−5,5√(𝑎−𝑏)(𝑎−𝑐),−5√(𝑎−𝑏)(𝑎−𝑐)
  • C−5,25√(𝑎−𝑏)(𝑎−𝑐),−25√(𝑎−𝑏)(𝑎−𝑐)
  • D−1,5√(𝑎−𝑏)(𝑎−𝑐),−5√(𝑎−𝑏)(𝑎−𝑐)
  • E−1,√(𝑎−𝑐)(5𝑎−5𝑏),−√(𝑎−𝑐)(5𝑎−5𝑏)

Q13:

Find, without expanding, the value of the determinant ||||180−𝑦𝑦−𝑧𝑧−180𝑦−𝑧𝑧−180180−𝑦𝑧−180180−𝑦𝑦−𝑧||||.

Q14:

Given that 𝑛=||||6−8915−9−11−72−4|||| and 𝑚=||||18−242790−54−66−3510−20||||, 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 ||||9−2𝑘−2𝑚7𝑛−2𝑘−1−2𝑚7𝑛−2𝑘−2𝑚9+7𝑛||||.

  • A−9(9+18𝑚+7𝑛+2𝑘)
  • B−9(9+18𝑚−7𝑛−2𝑘)
  • C−9(9+18𝑚+7𝑛−2𝑘)
  • D−81−18𝑚−7𝑛+18𝑘
  • E−9(9+18𝑚−7𝑛+2𝑘)

Q17:

Select a factor of the determinant||||𝑥−4−8𝑥−8𝑥+1𝑥+1548𝑥+8||||.

  • A𝑥+8
  • B𝑥
  • C𝑥+6
  • D𝑥+9

Q18:

Consider the equation ||||6−2011−14−1814−1612||||=298.

Find, without expanding, the value of the determinant ||||2−16114−14−18116−20||||.

Q19:

Put the determinant ||||−4−578−41652238|||| in upper triangular form, and find its value.

  • A||||−400−5−140730−9||||, −5,544
  • B||||−4−570−1430009||||, 504
  • C||||−4−57014−30009||||, −504
  • D||||−400−5−14073099||||, 5,544

Q20:

Use the properties of determinants to find ||||6𝑦+4𝑧5𝑥−4𝑧5𝑥4𝑧+5𝑥6𝑦−4𝑧6𝑦5𝑥+6𝑦04𝑧||||.

Q21:

Consider ||𝑥𝑦𝑧𝑤||=6.

Find the value of |||(𝑥−10𝑦)𝑦(𝑧−10𝑤)𝑤|||.

Q22:

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

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

Q23:

Use the properties of determinants to find the values of 𝑀=||||276−372−494||||,𝑀=||||−2296−6−12−50||||.

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

  • A𝑀=−2, 𝑀=−148, 𝑀𝑀=||||50−681152−58−3470−82−45||||
  • B𝑀=54, 𝑀=−156, 𝑀𝑀=||||−41454−18−42−2−8−450||||
  • C𝑀=54, 𝑀=−156, 𝑀𝑀=||||50−681152−58−3470−82−45||||
  • D𝑀=−2, 𝑀=−148, 𝑀𝑀=||||−41454−18−42−2−8−450||||

Q24:

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

Q25:

Using the properties of determinants, find the value of |||||3𝑥−1(𝑥−4)(𝑥−1)3𝑥+2(𝑥−1)(𝑥+2)3𝑥+5(𝑥+2)(𝑥+5)|||||.

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