Worksheet: Properties of Determinants

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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 | | | | − 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 0≤𝑥≤360∘∘.

  • 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+3−34−3𝑥−1−43−5𝑥+𝑘||||.

  • A 4 3
  • 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 ) ( 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 ||||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.

  • 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 ||||9−2𝑘−2𝑚7𝑛−2𝑘−1−2𝑚7𝑛−2𝑘−2𝑚9+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||||𝑥−4−8𝑥−8𝑥+1𝑥+1548𝑥+8||||.

  • A 𝑥 + 8
  • B 𝑥
  • C 𝑥 + 6
  • D 𝑥 + 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 | | | | − 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+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 ( 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=||||276−372−494||||,𝑀=||||−2296−6−12−50||||.

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 |||||3𝑥−1(𝑥−4)(𝑥−1)3𝑥+2(𝑥−1)(𝑥+2)3𝑥+5(𝑥+2)(𝑥+5)|||||.

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