Worksheet: Properties of Determinants

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In this worksheet, we will practice using the properties of determinants to solve problems.

Q1:

Find the value of | | | | 4 1 − 8 − 6 3 6 0 0 0 | | | | .

Q2:

Find the value of | | | | − 5 2 − 4 0 5 0 3 0 0 | | | | .

Q3:

Using the properties of determinants, find the value of | | | | − 1 3 3 4 − 8 − 7 0 3 − 5 | | | | .

Q4:

Use the properties of determinants to evaluate | | | | 1 1 1 4 7 − 1 4 − 5 7 − 3 9 1 4 | | | | .

Q5:

Without expanding, find the value of the determinant | | | | 8 − 3 − 2 7 1 − 8 2 4 − 9 − 6 | | | | .

Q6:

Using the properties of determinants, find the value of | | | | 5 0 𝑥 𝑥 5 0 𝑥 𝑥 5 0 𝑥 − 𝑥 | | | | . s e c s e c t a n s e c s e c t a n

Q7:

Evaluate | | 1 1 1 2 1 4 1 5 | | + | | 1 2 1 3 1 5 1 6 | | + | | 1 3 1 4 1 6 1 7 | | + ⋯ + | | 2 5 2 6 2 8 2 9 | | .

Q8:

Evaluate | | − 6 1 1 1 | | + | | − 5 1 1 1 | | + | | − 4 1 1 1 | | + ⋯ + | | 1 0 1 1 1 | | .

Q9:

Which of the following is equal to the determinant | | | | − 9 𝑏 9 𝑏 7 𝑐 2 𝑐 − 7 𝑐 − 6 𝑑 − 𝑎 6 𝑎 − 3 𝑏 | | | | ?

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

Q10:

Consider the equation | | | | 𝑎 𝑏 𝑐 𝑒 𝑓 𝑔 𝑥 𝑦 𝑧 | | | | = 2 9 .

Evaluate 𝑎 | | | 𝑏 𝑐 𝑓 𝑔 | | | − 𝑏 | | 𝑎 𝑐 𝑒 𝑔 | | + 𝑐 | | | 𝑎 𝑏 𝑒 𝑓 | | | .

Q11:

Consider the equation | | | | 1 0 0 5 2 𝑥 + √ 2 0 2 𝑚 1 + 2 𝑥 | | | | = 0 . s i n c o s

Determine all the possible values of 𝑥 given that 0 ≤ 𝑥 ≤ 3 6 0 ∘ ∘ .

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

Q12:

Use the properties of determinants to find the value of 𝑘 which makes 𝑥 a factor of the determinant | | | | 𝑥 + 3 − 3 4 − 3 𝑥 − 1 − 4 3 − 5 𝑥 + 𝑘 | | | | .

  • A36
  • B − 4
  • C 4 3
  • D4

Q13:

Consider the equation | | | | 5 𝑎 − 𝑥 5 𝑏 5 𝑐 5 𝑐 5 𝑏 − 𝑥 5 𝑎 5 𝑏 5 𝑎 5 𝑐 − 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 − 1 , 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 ) , − 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 )
  • C − 1 , √ ( 𝑎 − 𝑐 ) ( 5 𝑎 − 5 𝑏 ) , − √ ( 𝑎 − 𝑐 ) ( 5 𝑎 − 5 𝑏 )
  • D − 5 , 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 ) , − 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 )
  • E − 5 , 2 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 ) , − 2 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 )

Q14:

Find, without expanding, the value of the determinant | | | | 1 8 0 − 𝑦 𝑦 − 𝑧 𝑧 − 1 8 0 𝑦 − 𝑧 𝑧 − 1 8 0 1 8 0 − 𝑦 𝑧 − 1 8 0 1 8 0 − 𝑦 𝑦 − 𝑧 | | | | .

Q15:

Given that 𝑛 = | | | | 6 − 8 9 1 5 − 9 − 1 1 − 7 2 − 4 | | | | and 𝑚 = | | | | 1 8 − 2 4 2 7 9 0 − 5 4 − 6 6 − 3 5 1 0 − 2 0 | | | | , find a relation between 𝑚 and 𝑛 without expanding either determinant.

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

Q16:

Select the determinant that is equal to | | | | 2 3 2 4 9 4 2 3 2 2 5 2 8 2 1 1 4 | | | | .

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

Q17:

Find, in its simplest form, an expression for the determinant | | | | 9 − 2 𝑘 − 2 𝑚 7 𝑛 − 2 𝑘 − 1 − 2 𝑚 7 𝑛 − 2 𝑘 − 2 𝑚 9 + 7 𝑛 | | | | .

  • 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 − 9 ( 9 + 1 8 𝑚 + 7 𝑛 − 2 𝑘 )
  • E − 8 1 − 1 8 𝑚 − 7 𝑛 + 1 8 𝑘

Q18:

Select a factor of the determinant | | | | 𝑥 − 4 − 8 𝑥 − 8 𝑥 + 1 𝑥 + 1 5 4 8 𝑥 + 8 | | | | .

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

Q19:

Consider the equation | | | | 6 − 2 0 1 1 − 1 4 − 1 8 1 4 − 1 6 1 2 | | | | = 2 9 8 .

Find, without expanding, the value of the determinant | | | | 2 − 1 6 1 1 4 − 1 4 − 1 8 1 1 6 − 2 0 | | | | .

Q20:

Put the determinant | | | | − 4 − 5 7 8 − 4 1 6 5 2 2 3 8 | | | | 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 | | | | , − 5 0 4
  • C | | | | − 4 0 0 − 5 − 1 4 0 7 3 0 9 9 | | | | , 5‎ ‎544
  • D | | | | − 4 − 5 7 0 − 1 4 3 0 0 0 9 | | | | , 504

Q21:

Which of the following pairs of points does the straight line represented by the equation | | | | 𝑦 𝑥 − 8 3 − 9 − 5 0 − 1 − 1 | | | | = 0 pass through?

  • A ( 0 , − 6 ) , ( − 3 , − 4 )
  • B ( − 4 , − 3 ) , ( − 6 , 0 )
  • C ( − 3 , − 4 ) , ( − 6 , 0 )
  • D ( − 4 , − 3 ) , ( 0 , − 6 )

Q22:

Use the properties of determinants to find | | | | 6 𝑦 + 4 𝑧 5 𝑥 − 4 𝑧 5 𝑥 4 𝑧 + 5 𝑥 6 𝑦 − 4 𝑧 6 𝑦 5 𝑥 + 6 𝑦 0 4 𝑧 | | | | .

Q23:

Consider | | 𝑥 𝑦 𝑧 𝑤 | | = 6 .

Find the value of | | | ( 𝑥 − 1 0 𝑦 ) 𝑦 ( 𝑧 − 1 0 𝑤 ) 𝑤 | | | .

Q24:

Find the solution set of the equation | | − 8 𝑥 − 1 − 7 5 𝑥 | | + | | − 7 1 0 − 4 − 7 𝑥 | | = 9 .

  • A − 1 , 3 5
  • B − 8 5 , 3 8
  • C − 3 5 , 1
  • D − 3 8 , 8 5
  • E − 1 5 , 3

Q25:

Find, in its simplest form, an expression for the determinant | | | | 7 𝑦 3 𝑥 3 𝑥 3 𝑥 7 𝑦 3 𝑥 3 𝑥 3 𝑥 7 𝑦 | | | | .

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

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