Worksheet: Two-by-Two Determinants

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In this worksheet, we will practice identifying determinants and evaluating 2x2 determinants.

Q1:

Find all the values of 𝑥 for which | | 𝑥 − 2 − 2 𝑥 | | + | | 6 3 1 8 | | = 4 5 .

  • A1 or − 1
  • B4
  • C4 or − 4
  • D2 or − 2
  • E1

Q2:

Solve | | 𝑥 5 4 − 9 | | = 1 6 .

  • A 𝑥 = − 2 0 9
  • B 𝑥 = 3 6
  • C 𝑥 = 4 9
  • D 𝑥 = − 4

Q3:

Solve | | 𝑥 2 − 5 𝑥 | | − | | − 5 2 3 𝑥 | | = 1 0 .

  • A − 5 or − 2
  • B − 2 or − 3
  • C2 or 3
  • D − 1 or 3
  • E2 or 1

Q4:

Find the value of | | | 𝑎 + 𝑥 5 𝑎 𝑏 + 7 𝑦 − 7 𝑏 | | | .

  • A − 7 𝑏 𝑥 − 3 5 𝑎 𝑦 − 2 𝑎 𝑏
  • B − 7 𝑏 𝑥 − 3 5 𝑎 𝑦 − 1 2 𝑎 𝑏
  • C − 7 𝑏 𝑥 − 3 5 𝑎 𝑦
  • D − 7 𝑏 𝑥 + 3 5 𝑎 𝑦 − 2 𝑎 𝑏
  • E − 7 𝑏 𝑥 − 1 2 𝑎 𝑏

Q5:

Find the value of the determinant | | | 𝜃 𝜃 − 𝜃 𝜃 | | | c o t t a n c o t t a n .

Q6:

Find the value of | | | − 3 𝑥 + 3 4 𝑥 + 4 − 3 𝑦 + 4 − 𝑦 + 3 | | | .

  • A 3 𝑥 𝑦 − 3 𝑥 − 𝑦 + 1 2 𝑥 𝑦 − 4 𝑥 + 3 𝑦
  • B 1 5 𝑥 𝑦 − 9 𝑥 − 3 𝑦 − 1 6 𝑥 + 1 2 𝑦 − 7
  • C 3 𝑥 𝑦 − 9 𝑥 − 3 𝑦 + 1 2 𝑥 𝑦 − 1 6 𝑥 + 1 2 𝑦 − 7
  • D 1 5 𝑥 𝑦 − 2 5 𝑥 + 9 𝑦 + 1 2 𝑥 𝑦 − 7
  • E 3 𝑥 𝑦 − 2 5 𝑥 + 9 𝑦 + 1 2 𝑥 𝑦 − 7

Q7:

Evaluate | | | − 4 8 𝜃 − 𝜃 2 𝜃 | | | . s e c s e c t a n

Q8:

Find all the values of 𝑥 for which | | 𝑥 − 5 − 5 𝑥 | | + | | 1 − 2 4 8 | | = 4 0 .

  • A49
  • B49 or − 4 9
  • C64
  • D7 or − 7
  • E8 or − 8

Q9:

Solve | | 𝑥 3 − 4 𝑥 | | + | | 6 2 − 4 𝑥 | | = 1 2 .

  • A4 or 1
  • B4 or 2
  • C − 4 or − 2
  • D − 1 or 2
  • E − 4 or − 3

Q10:

Solve | | 𝑥 3 1 0 − 5 | | = − 4 0 .

  • A 𝑥 = 1 4
  • B 𝑥 = 2
  • C 𝑥 = − 7 0
  • D 𝑥 = − 6
  • E 𝑥 = − 1 0

Q11:

Find the value of the determinant 𝐴 = | | 𝑥 − 1 1 𝑥 − 1 | | .

  • A 1 2 𝑥
  • B − 1 0 𝑥
  • C − 𝑥
  • D − 1 2 𝑥
  • E 1 0 𝑥

Q12:

Find the value of 𝑥 which satisfies | | 3 4 0 − 2 5 0 − 3 4 0 2 5 0 | | = 5 𝑥 − 4 s i n s i n c o s c o s ∘ ∘ ∘ ∘ .

  • A1
  • B − 4 5
  • C − 1
  • D 3 5

Q13:

Use the properties of determinants to evaluate | | 7 6 0 7 6 1 3 3 5 3 3 6 | | .

Q14:

Evaluate | | 3 − 4 1 5 | | .

  • A − 1 1
  • B − 1 9
  • C19
  • D11
  • E − 7

Q15:

Find the value of | | − 2 − 9 7 7 | | .

Q16:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 3 5 − 1 2 𝑥 𝑦 = 2 3 − 4

  • A 2 3 𝑥 − 4 𝑦 = 3 − 4 𝑥 + 1 7 𝑦 = − 1
  • B 3 𝑥 + 5 𝑥 = 2 3 − 𝑦 + 2 𝑦 = − 4
  • C 3 𝑥 + 5 𝑦 = 2 3 − 𝑥 + 2 𝑦 = − 4
  • D 3 𝑦 + 5 𝑥 = 2 3 − 𝑦 + 2 𝑥 = − 4
  • E 3 𝑥 − 𝑦 = 2 3 5 𝑥 + 2 𝑦 = − 4

Q17:

Solve the system of the linear equations 3 𝑥 + 2 𝑦 = 8 and 6 𝑥 − 9 𝑦 = 3 using the inverse of a matrix.

  • A 𝑥 = 0 , 𝑦 = 1 3
  • B 𝑥 = 1 3 , 𝑦 = 0
  • C 𝑥 = 2 , 𝑦 = 1
  • D 𝑥 = 1 , 𝑦 = 2

Q18:

Consider the simultaneous equations 3 𝑥 + 2 𝑦 = 1 2 3 𝑥 + 𝑦 = 7 .

Express the simultaneous equations as a single matrix equation.

  • A 3 2 3 1 𝑥 𝑦 = 7 1 2
  • B 3 3 2 1 𝑥 𝑦 = 7 1 2
  • C 2 3 1 3 𝑥 𝑦 = 7 1 2
  • D 2 3 1 3 𝑥 𝑦 = 1 2 7
  • E 3 2 3 1 𝑥 𝑦 = 1 2 7

Write down the inverse of the coefficient matrix.

  • A 1 3 1 − 2 − 3 3
  • B 1 3 − 1 3 2 − 3
  • C 1 9 1 3 2 3
  • D 1 3 − 1 2 3 − 3
  • E 1 9 1 − 2 − 3 3

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑥 𝑦 = 2 3 5
  • B 𝑥 𝑦 = 1 9 8 2 3
  • C 𝑥 𝑦 = − 5 2 3 5
  • D 𝑥 𝑦 = − 5 5 2 3
  • E 𝑥 𝑦 = 9 2 3 − 7 1 3

Q19:

Using matrix inverses, solve the following for 𝑋 : 𝑋 − 3 2 4 − 3 = 0 − 2 3 0 .

  • A 𝑋 = 8 6 9 6
  • B 𝑋 = 8 − 6 9 − 6
  • C 𝑋 = − 8 6 − 9 − 6
  • D 𝑋 = 6 6 − 9 − 8
  • E 𝑋 = 8 6 − 9 − 6

Q20:

Use matrices to solve the system − 9 𝑥 + 3 𝑦 − 5 = 0 , 5 𝑥 = − 3 + 5 𝑦 .

  • A 𝑥 = − 4 1 5 , 𝑦 = 1 3 0
  • B 𝑥 = 1 1 5 , 𝑦 = − 8 1 5
  • C 𝑥 = 5 , 𝑦 = 5 0 3
  • D 𝑥 = 8 1 5 , 𝑦 = − 1 1 5
  • E 𝑥 = − 8 1 5 , 𝑦 = 1 1 5

Q21:

Use matrices to solve the system of equations 3 𝑥 − 2 4 = − 8 𝑦 , 𝑥 = 3 − 𝑦 .

  • A 𝑥 𝑦 = − 3 6
  • B 𝑥 𝑦 = 4 − 1
  • C 𝑥 𝑦 = 6 − 3
  • D 𝑥 𝑦 = 0 3
  • E 𝑥 𝑦 = 3 0

Q22:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 3 3 2 4 𝑎 𝑏 = 1 0 1 2

  • A 3 𝑎 + 3 𝑏 = 1 0 4 𝑎 + 2 𝑏 = 1 2
  • B 3 𝑎 + 2 𝑏 = 1 2 3 𝑎 + 4 𝑏 = 1 0
  • C 3 𝑎 + 3 𝑏 = 1 0 2 𝑎 + 4 𝑏 = 1 2
  • D 3 𝑎 + 3 𝑏 = 1 2 2 𝑎 + 4 𝑏 = 1 0
  • E 3 𝑎 + 2 𝑏 = 1 0 3 𝑎 + 4 𝑏 = 1 2

Q23:

Consider the system of equations 3 𝑎 + 2 𝑏 = 1 3 2 𝑎 + 3 𝑏 = 7 .

Express the system as a single matrix equation.

  • A 3 2 3 2 𝑎 𝑏 = 1 3 7
  • B 3 3 2 2 𝑎 𝑏 = 1 3 7
  • C 3 3 2 2 𝑎 𝑏 = 7 1 3
  • D 3 2 2 3 𝑎 𝑏 = 1 3 7
  • E 3 2 2 3 𝑎 𝑏 = 7 1 3

Write down the inverse of the coefficient matrix.

  • A 1 1 2 3 − 2 − 2 3
  • B 1 5 3 − 2 − 2 3
  • C 1 5 − 3 2 2 − 3
  • D 1 5 3 2 2 3
  • E 1 1 2 2 − 3 − 2 3

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑎 𝑏 = 1 − 5
  • B 𝑎 𝑏 = ⎡ ⎢ ⎢ ⎣ 5 3 5 4 7 5 ⎤ ⎥ ⎥ ⎦
  • C 𝑎 𝑏 = ⎡ ⎢ ⎢ ⎣ − 2 5 1 2 2 5 1 2 ⎤ ⎥ ⎥ ⎦
  • D 𝑎 𝑏 = ⎡ ⎢ ⎢ ⎣ 4 7 1 2 4 7 1 2 ⎤ ⎥ ⎥ ⎦
  • E 𝑎 𝑏 = 5 − 1

Q24:

Given 𝐴 1 0 2 1 0 1 = 1 0 0 1 , find the matrix 𝐴 .

  • A 1 2 1 0 1 0
  • B 1 − 2 − 1 0 1 0
  • C ⎡ ⎢ ⎢ ⎣ − 1 − 1 5 − 1 − 1 1 0 ⎤ ⎥ ⎥ ⎦
  • D − 1 1 0 1 5 1 − 1

Q25:

Given that the solution set of the equation 𝑎 𝑥 + 𝑏 𝑥 + 7 = 0 is { − 1 , − 7 } , use matrices to find the constants 𝑎 and 𝑏 .

  • A 𝑎 = 8 , 𝑏 = 1
  • B 𝑎 = − 1 , 𝑏 = − 8
  • C 𝑎 = 1 , 𝑏 = 8
  • D 𝑎 = − 3 4 , 𝑏 = − 6

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