Worksheet: Symmetric and Skew-Symmetric Matrices

In this worksheet, we will practice determining whether a matrix is symmetric or skew-symmetric and using their properties to solve problems.

Q1:

Which of the following matrices is skew-symmetric?

  • A 3 5 2 5 3 1 2 1 3
  • B 1 0 9 2 9 2 5 2 5 4
  • C 0 1 9 1 0 6 9 6 7
  • D 0 3 5 3 0 1 0 5 1 0 0

Q2:

If is a symmetric matrix, what are the values of 𝑥 , 𝑦 , and 𝑧 ?

  • A 𝑥 = 9 , 𝑦 = 0 , 𝑧 = 5
  • B 𝑥 = 9 , 𝑦 = 0 , 𝑧 = 1 7
  • C 𝑥 = 9 , 𝑦 = 1 2 , 𝑧 = 1 7
  • D 𝑥 = 9 , 𝑦 = 0 , 𝑧 = 5
  • E 𝑥 = 6 , 𝑦 = 0 , 𝑧 = 5

Q3:

Which of the following matrices is symmetric?

  • A 3 5 1 6 6 5 7 7 8
  • B 4 4 2 4 8 2 8 2 3
  • C 2 4 3 4 8 8 9 3 2
  • D 2 3 5 3 4 3 5 3 7

Q4:

Find the value of 𝑥 which makes the matrix 𝐴 = 1 5 𝑥 3 4 3 8 symmetric.

Q5:

Suppose that 𝐴 and 𝐵 are symmetric matrices and that 𝐴 𝐵 = 𝐵 𝐴 . Does it follow that 𝐴 𝐵 is also a symmetric matrix?

  • Ayes
  • Bno

Q6:

If 𝐴 and 𝐵 are symmetric matrices, does it follow that 𝐴 𝐵 is also symmetric?

  • Ano
  • Byes

Q7:

Given that the matrix is skew-symmetric, find the value of 𝑥 + 𝑦 + 𝑧 .

Q8:

The non-zero eigenvalues of a real, skew-symmetric matrix are .

  • Apurely real
  • Bnon-negative
  • Cpurely imaginary

Q9:

Suppose 𝐴 is a 4 × 4 real matrix. Which of the following is true about the matrix 𝐴 𝐴 ?

  • Ais always singular
  • Bis always skew-symmetric
  • Cis always nonsingular
  • Dis always symmetric

Q10:

If is a symmetric matrix, what are the values of 𝑥 , 𝑦 , and 𝑧 ?

  • A 𝑥 = 6 , 𝑦 = 0 , 𝑧 = 1
  • B 𝑥 = 2 , 𝑦 = 0 , 𝑧 = 9
  • C 𝑥 = 2 , 𝑦 = 1 0 , 𝑧 = 9
  • D 𝑥 = 2 , 𝑦 = 0 , 𝑧 = 1
  • E 𝑥 = 3 , 𝑦 = 0 , 𝑧 = 1

Q11:

Which of the following matrices is skew-symmetric?

  • A 3 1 8 1 1 1 8 1 4
  • B 9 6 7 6 2 3 7 3 3
  • C 0 4 4 4 0 6 4 6 9
  • D 0 3 4 3 0 8 4 8 0

Q12:

Which of the following matrices is skew-symmetric?

  • A 2 2 8 2 7 1 0 8 1 0 1 0
  • B 2 8 3 8 3 5 3 5 8
  • C 0 1 0 7 1 0 0 8 7 8 6
  • D 0 4 9 4 0 3 9 3 0

Q13:

Given that the matrix is skew-symmetric, find the value of 𝑥 + 𝑦 + 𝑧 .

Q14:

Given that the matrix is skew-symmetric, find the value of 𝑥 + 𝑦 + 𝑧 .

Q15:

Find the value of 𝑥 which makes the matrix 𝐴 = 9 7 𝑥 5 5 8 4 symmetric.

Q16:

Find the value of 𝑥 which makes the matrix 𝐴 = 6 𝑥 + 9 3 8 symmetric.

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