Worksheet: Invertible and Singular Matrices

In this worksheet, we will practice differentiating between invertible and singular matrices.

Q1:

Is the following matrix invertible?

  • Ano
  • Byes

Q2:

Under what condition on is the following matrix invertible?

  • A is any real number
  • B or
  • C and
  • D and
  • E or

Q3:

Is there any value of for which the matrix has no inverse?

  • Ayes, when
  • Byes, when
  • Cyes, when
  • Dno
  • Eyes, when

Q4:

Is there any value of for which the matrix has no inverse?

  • Ayes, when
  • Byes, when
  • Cno
  • Dyes, the matrix has no inverse for all values of
  • Eyes, when

Q5:

Is there any value of for which the matrix has no inverse?

  • A yes, when
  • B yes, when
  • C yes, when
  • D no
  • E yes, when

Q6:

Which of the following matrices is singular?

  • A 9 7 9 7
  • B 9 7 9 7
  • C 9 7 9 7
  • D 9 7 9 7

Q7:

Under what condition on 𝑎 and 𝑏 is the following matrix invertible?

  • A 2 𝑎 + 5 𝑏 0
  • B 2 𝑏 = 5 𝑎
  • C 5 𝑏 2 𝑎
  • D 2 𝑏 5 𝑎
  • E 𝑎 𝑏

Q8:

Find the value of 𝑥 that makes the matrix 𝑥 8 5 1 singular.

  • A 1 4 0
  • B40
  • C 1 4 0
  • D 4 0

Q9:

Find all the values of 𝑥 for which the matrix 𝑥 1 1 5 1 5 𝑥 + 1 1 is singular.

  • A4
  • B 5 , 5
  • C196
  • D 1 4 , 1 4

Q10:

Find the value of 𝑥 that makes the matrix singular.

Q11:

Find all the values of 𝑥 , where 0 𝑥 < 3 6 0 , which make the following matrix singular:

  • A 9 0 , 2 7 0
  • B 3 0 , 2 1 0
  • C 1 2 0 , 3 0 0
  • D 0 , 1 8 0

Q12:

Find the set of real values of 𝑎 for which has a multiplicative inverse.

  • A
  • B { 1 1 , 3 }
  • C { 7 , 1 }
  • D { 1 1 , 3 }
  • E { 4 , 1 0 }

Q13:

For the matrix does there exist a value of for which it fails to have an inverse? if so, what is this value?

  • A yes, when .
  • B yes, when .
  • C yes, when .
  • D yes, when .
  • E yes, when .

Q14:

Find the set of real values of 𝑥 for which 4 𝑥 6 8 1 7 𝑥 has no multiplicative inverse.

  • A { 1 7 }
  • B 1 7 4 , 1 7 4
  • C 1 7 4
  • D { 1 7 , 1 7 }
  • E { 1 7 }

Q15:

Find the set of real values of 𝑎 for which 𝑎 8 8 𝑎 has no multiplicative inverse.

  • A { 8 }
  • B { 6 4 , 6 4 }
  • C { 6 4 }
  • D { 8 , 8 }

Q16:

Find the set of real values of 𝑎 for which the following matrix has a multiplicative inverse

  • A
  • B { 2 6 }
  • C { 2 6 }
  • D { 2 6 }
  • E { 2 6 }

Q17:

Find the set of real values of 𝑎 for which 𝑎 𝑖 𝑖 1 2 has a multiplicative inverse, where 𝑖 = 1 2 .

  • A { 1 }
  • B { 1 }
  • C { 0 }
  • D

Q18:

Does the matrix have a multiplicative inverse?

  • ANo
  • BYes

Q19:

Find the set of real values of ( 𝑥 ) that make the following matrix singular.

  • A { 6 , 4 }
  • B { 4 , 6 }
  • C { 7 , 7 }
  • D { 5 , 5 }

Q20:

Find the set of real values of 𝑥 that make the matrix singular.

  • A { 8 , 6 }
  • B { 6 , 8 }
  • C { 5 , 5 }
  • D { 7 , 7 }

Q21:

Suppose 𝐴 is matrix in × and v is a vector in and consider this statement: there exists a solution x to the equation 𝐴 = x v . Which of the following is true?

  • AThe statement is true if and only if 𝐴 is nonsingular.
  • B If the statement is true, then 𝐴 is nonsingular.
  • CThe statement is always true.
  • DIf 𝐴 is nonsingular, then the statement is true.
  • E If 𝐴 is nonsingular, then the statement is false.

Q22:

Which of the following matrices is singular?

  • A 3 7 3 7
  • B 3 7 3 7
  • C 3 7 3 7
  • D 3 7 3 7

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