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
• B
• C
• D

Q2:

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

• A , ,
• B , ,
• C , ,
• D , ,
• E , ,

Q3:

Which of the following matrices is symmetric?

• A
• B
• C
• D

Q4:

Find the value of which makes the matrix symmetric.

Q5:

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

• Ayes
• Bno

Q6:

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

• Ano
• Byes

Q7:

Given that the matrix is skew-symmetric, find the value of .

Q8:

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

• Apurely real
• Bnon-negative
• Cpurely imaginary

Q9:

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

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

Q10:

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

• A , ,
• B , ,
• C , ,
• D , ,
• E , ,

Q11:

Which of the following matrices is skew-symmetric?

• A
• B
• C
• D

Q12:

Which of the following matrices is skew-symmetric?

• A
• B
• C
• D

Q13:

Given that the matrix is skew-symmetric, find the value of .

Q14:

Given that the matrix is skew-symmetric, find the value of .

Q15:

Find the value of which makes the matrix symmetric.

Q16:

Find the value of which makes the matrix symmetric.