Mathematics Questions and Answers – Symmetric and Skew Symmetric Matrices

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Symmetric and Skew Symmetric Matrices”.

1. The matrix A=\(\begin{bmatrix}1&2\\2&1\end{bmatrix}\) is a ____________
a) symmetric matrix
b) skew-symmetric matrix
c) null matrix
d) diagonal matrix
View Answer

Answer: a
Explanation: Given that, A=\(\begin{bmatrix}1&2\\2&1\end{bmatrix}\)
⇒ A’=\(\begin{bmatrix}1&2\\2&1\end{bmatrix}\)
i.e.A=A’. Hence, it is a symmetric matrix.
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2. Which of the following conditions holds true for a symmetric matrix?
a) A=-A’
b) A=A’
c) A=IA
d) A=|A|
View Answer

Answer: b
Explanation: A matrix is A said to be a symmetric matrix if it is equal to its transpose i.e. A=A’.

3. Which of the following conditions holds true for a skew-symmetric matrix?
a) A=IA
b) A=|A|
c) A=A’
d) A=-A’
View Answer

Answer: a
Explanation: A matrix is said to be skew-symmetric if it is equal to the negative of its transpose i.e. A=-A’.
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4. Any square matrix can be expressed as a sum of symmetric and skew-symmetric matrix.
a) True
b) False
View Answer

Answer: a
Explanation: The given statement is true. Every square matrix can be expressed as a sum of sum of symmetric and skew-symmetric matrix.
If A is a square matrix then it can be expressed as
A = \(\frac{1}{2}\)(A+A’)+\(\frac{1}{2}\)(A-A’), where (A+A’) is symmetric and (A-A’) is skew-symmetric.

5. The matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is __________
a) scalar matrix
b) identity matrix
c) symmetric matrix
d) skew-symmetric matrix
View Answer

Answer: d
Explanation: The given matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is skew symmetric.
⇒A’=\(\begin{bmatrix}0&-1&1\\1&0&-1\\-1&1&0\end{bmatrix}\)=A
∴A=-A’. Hence, it is a skew-symmetric matrix.
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6. Which of the following matrices is both symmetric and skew symmetric?
a) A=\(\begin{bmatrix}1&0\\1&0\end{bmatrix}\)
b) A=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)
c) A=\(\begin{bmatrix}1&0&1\\1&0&1\end{bmatrix}\)
d) A=\(\begin{bmatrix}0&0&-2\\1&0&-1\\2&0&0\end{bmatrix}\)
View Answer

Answer: b
Explanation: The matrix A=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)=A’=-A’.Hence, a null matrix is both symmetric and skew-symmetric.

7. The matrix A=\(\begin{bmatrix}0&1&1\\1&0&-1\\-1&1&0\end{bmatrix}\) is symmetric.
a) True
b) False
View Answer

Answer: b
Explanation: Given that, A=\(\begin{bmatrix}0&1&1\\1&0&-1\\-1&1&0\end{bmatrix}\)
⇒A’=\(\begin{bmatrix}0&1&-1\\1&0&1\\1&-1&0\end{bmatrix}\). ∴A ≠ A’. Hence, it is not symmetric.
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8. The matrix A=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\) as a sum of symmetric and skew-symmetric matrix is ______
a) \( \frac{1}{4} \begin{bmatrix}4&11\\11&12\end{bmatrix} – \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)
b) \( \frac{1}{4} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\7&0\end{bmatrix}\)
c) \( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)
d) \( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} – \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)
View Answer

Answer: c
Explanation: Given that A=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\).
A’=\(\begin{bmatrix}2&2\\9&6\end{bmatrix}\)
⇒A+A’=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\)+\(\begin{bmatrix}2&2\\9&6\end{bmatrix}\)=\(\begin{bmatrix}4&11\\11&12\end{bmatrix}\)
⇒A-A’=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\)–\(\begin{bmatrix}2&2\\9&6\end{bmatrix}\)=\(\begin{bmatrix}0&7\\-7&0\end{bmatrix}\)
The given square matrix can be written as
⇒A = \( \frac{1}{2}\) (A+A’) + \( \frac{1}{2}\) (A-A’)=\( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\).

9. If A=\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\), then which of the following statement is incorrect?
a) A is a skew-symmetric matrix
b) A is a square matrix
c) A is a symmetric
d) A is an identity matrix
View Answer

Answer: a
Explanation: Given that, A=\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
∴A’=\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
⇒-A’=\(\begin{bmatrix}-1&0\\0&-1\end{bmatrix}\)≠A. Hence, it is not a skew symmetric matrix.
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10. If A=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), then which of the following is skew-symmetric?
a) AA’
b) A+A’
c) 2(A+A’)
d) A-A’
View Answer

Answer: c
Explanation: Given that, A=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\)
⇒A’=\(\begin{bmatrix}a&c\\b&d\end{bmatrix}\)
Let B=A-A’=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\)–\(\begin{bmatrix}a&c\\b&d\end{bmatrix}\)=\(\begin{bmatrix}a-a&b-c\\c-b&d-d\end{bmatrix}\)=\(\begin{bmatrix}0&b-c\\c-b&0\end{bmatrix}\)
B’=\(\begin{bmatrix}0&c-b\\b-c&0\end{bmatrix}\)=B’
Thus, B=A-A’ is a skew – symmetric.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter