Worksheet: Three-by-Three Determinants

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In this worksheet, we will practice evaluating 3x3 determinants using the cofactors (Laplace expansion) or the Sarrus method.

Q1:

Find the value of ||||βˆ’25βˆ’851800βˆ’5||||.

Q2:

Consider the determinant ||||π‘₯𝑧𝑦𝑦π‘₯𝑧𝑧𝑦π‘₯||||. Given that π‘₯+𝑦+𝑧=βˆ’73 and π‘₯𝑦𝑧=βˆ’8, determine the determinant’s numerical value.

Q3:

Which of the following is equal to the determinant ||||π‘βˆ’8π‘π‘βˆ’7π‘Žπ‘Žβˆ’7𝑏8𝑐7π‘Ž7π‘βˆ’6βˆ’6βˆ’6||||?

  • A 6 ( π‘Ž βˆ’ 𝑐 ) ( 7 𝑏 βˆ’ 8 𝑐 ) + 4 2 ( π‘Ž βˆ’ 𝑏 ) 
  • B 6 ( π‘Ž βˆ’ 𝑐 ) ( 7 𝑏 βˆ’ 8 𝑐 ) βˆ’ 7 ( π‘Ž βˆ’ 𝑏 ) 
  • C 6 ( π‘Ž βˆ’ 𝑐 ) ( 7 𝑏 βˆ’ 8 𝑐 ) βˆ’ 4 2 ( π‘Ž βˆ’ 𝑏 ) 
  • D βˆ’ 4 2 ( π‘Ž βˆ’ 𝑏 ) βˆ’ 6 ( π‘Ž βˆ’ 𝑐 ) ( 7 𝑏 βˆ’ 8 𝑐 ) 

Q4:

Evaluate ||||70𝑖+1019π‘–βˆ’π‘–+1βˆ’4π‘–βˆ’10||||.

Q5:

Given that πœ” is a complex cube root of unity, evaluate ||||βˆ’9πœ”πœ”77πœ”πœ”πœ”1||||.

Q6:

Evaluate the shown determinant ||||𝑖01+𝑖1+𝑖𝑖01βˆ’π‘–βˆ’π‘–π‘–||||, where 𝑖=βˆ’1.

Q7:

Evaluate||||1βˆ’9βˆ’6βˆ’8412βˆ’19||||.

Q8:

Find the value of ||||226βˆ’31βˆ’2βˆ’5βˆ’1βˆ’4||||.

Q9:

Find, in its simplest form, an expression for the determinant ||||5π‘₯βˆ’3π‘Žβˆ’6π‘βˆ’3π‘Ž5π‘₯βˆ’6π‘βˆ’3π‘Žβˆ’6𝑏5π‘₯||||.

  • A ( 5 π‘₯ + 3 π‘Ž ) ( 5 π‘₯ + 6 𝑏 )
  • B ( 5 π‘₯ βˆ’ 3 π‘Ž ) ( 5 π‘₯ βˆ’ 6 𝑏 ) ( 5 π‘₯ βˆ’ 3 π‘Ž βˆ’ 6 𝑏 )
  • C ( 5 π‘₯ + 3 π‘Ž ) ( 5 π‘₯ + 6 𝑏 ) ( 5 π‘₯ βˆ’ 3 π‘Ž βˆ’ 6 𝑏 )
  • D ( 5 π‘₯ + 3 π‘Ž ) ( 5 π‘₯ + 6 𝑏 ) ( 5 π‘₯ + 3 π‘Ž + 6 𝑏 )

Q10:

Calculate |𝐴| when 𝐴=⎑⎒⎒⎣30βˆ’1010224⎀βŽ₯βŽ₯⎦.

Q11:

Find the determinant of the matrix ⎑⎒⎒⎣1232132241501212⎀βŽ₯βŽ₯⎦.

Q12:

Find the determinant of the matrix 22+2𝑖3βˆ’3𝑖2βˆ’2𝑖51βˆ’7𝑖3+3𝑖1+7𝑖16ο₯.

Q13:

Calculate |𝐴| when 𝐴=⎑⎒⎒⎒⎒⎣14βˆ’50022βˆ’3βˆ’11412βˆ’300⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.

Q14:

Find the determinant of the matrix 123322098.

Q15:

Consider 𝐴=ο˜βˆ’65βˆ’326βˆ’899βˆ’7.

Write the determinant whose value is equal to the minor of the element π‘ŽοŠ¨οŠ©.

  • A | | βˆ’ 6 5 9 9 | |
  • B | | 6 βˆ’ 8 9 βˆ’ 7 | |
  • C | | βˆ’ 6 βˆ’ 3 2 βˆ’ 8 | |
  • D | | 5 βˆ’ 3 9 βˆ’ 7 | |

Q16:

Find the determinant of the matrix 102+6𝑖8βˆ’6𝑖2βˆ’6𝑖91βˆ’7𝑖8+6𝑖1+7𝑖17ο₯.

Q17:

Calculate the determinant of 𝐴=1βˆ’13246βˆ’2βˆ’31.

Q18:

Find the solution set of||||π‘₯00βˆ’1βˆ’5π‘₯021π‘₯||||=βˆ’80π‘₯.

  • A { 4 , βˆ’ 4 }
  • B  0 , 1 4 , βˆ’ 1 4 
  • C { 0 , 4 }
  • D  1 4 , βˆ’ 1 4 
  • E { 0 , 4 , βˆ’ 4 }

Q19:

Given that π‘Ž is a fixed constant, find solution set of the equation ||||βˆ’π‘Žβˆ’π‘₯5π‘Ž+π‘₯βˆ’3π‘Ž+π‘₯βˆ’π‘Ž+π‘₯5π‘Žβˆ’π‘₯βˆ’3π‘Ž+π‘₯βˆ’π‘Ž+π‘₯5π‘Ž+π‘₯βˆ’3π‘Žβˆ’π‘₯||||=0.

  • A { 0 , βˆ’ π‘Ž }
  • B { βˆ’ π‘Ž }
  • C { 0 , βˆ’ 9 π‘Ž }
  • D { βˆ’ 9 π‘Ž }

Q20:

Solve for π‘₯: ||||05βˆ’5π‘₯π‘₯45413||||=280.

  • A π‘₯ = 9 or π‘₯=4
  • B π‘₯ = βˆ’ 9 or π‘₯=4
  • C π‘₯ = βˆ’ 9 or π‘₯=βˆ’20
  • D π‘₯ = 9 or π‘₯=20

Q21:

Solve||βˆ’3π‘₯63βˆ’4||=2||||4βˆ’46βˆ’33βˆ’223π‘₯5||||.

  • A π‘₯ = 1 1 8 4
  • B π‘₯ = βˆ’ 1 1 8 4
  • C π‘₯ = βˆ’ 1 2 1
  • D π‘₯ = 7 7 3 6
  • E π‘₯ = βˆ’ 1 1 3 6

Q22:

Find the solution set of ||||βˆ’8π‘₯7π‘₯3π‘₯βˆ’2π‘₯0βˆ’5π‘₯βˆ’8π‘₯9π‘₯3π‘₯||||=736 in ℝ.

  • A { 2 }
  • B { βˆ’ 1 }
  • C { βˆ’ 3 }
  • D { βˆ’ 2 }

Q23:

Solve the equation ||||βˆ’8π‘Ž+π‘₯𝑏4𝑐4𝑐𝑏+π‘₯βˆ’8π‘Žβˆ’8π‘Žπ‘4𝑐+π‘₯||||=0.

  • A  Β± √ 8 π‘Ž + 4 𝑐 , 8 π‘Ž + 𝑏 βˆ’ 4 𝑐 
  • B { 0 , 8 π‘Ž + 𝑏 βˆ’ 4 𝑐 }
  • C { 0 , 8 π‘Ž βˆ’ 𝑏 βˆ’ 4 𝑐 }
  • D  Β± √ 8 π‘Ž + 4 𝑐 , 8 π‘Ž βˆ’ 𝑏 βˆ’ 4 𝑐 

Q24:

Determine the value of π‘˜ that makes π‘₯=4 a root of the equation ||||9π‘₯βˆ’3βˆ’4π‘₯+8βˆ’2π‘˜9π‘₯βˆ’3π‘₯βˆ’7βˆ’5||||=0.

  • A βˆ’ 1 5
  • B 3 2
  • C5
  • D βˆ’ 1 2

Q25:

Find the solution set of the equation ||||π‘₯003π‘₯π‘₯βˆ’44π‘₯||||=βˆ’3π‘₯.

  • A { 3 , 1 , 0 }
  • B { βˆ’ 3 , βˆ’ 1 , 0 }
  • C { βˆ’ 3 , βˆ’ 1 }
  • D  βˆ’ 1 3 , βˆ’ 1 , 0 

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