Worksheet: Properties of Determinants

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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