Symmetric, Skew-Symmetric & Orthogonal Matrices – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

Symmetric, Skew-Symmetric, and Orthogonal matrices are important concepts in Linear Algebra under Engineering Mathematics. These topics are frequently asked in GATE, ESE, AE/JE exams.

Basic Concepts

A matrix is classified based on its relation with its transpose.

Symmetric Matrix

A matrix A is said to be symmetric if:

A^T = A

Example

A =

A^T = A → Symmetric Matrix

Skew-Symmetric Matrix

A matrix A is said to be skew-symmetric if:

A^T = −A

Important Property

• All diagonal elements are zero

Example

A =

A^T = −A → Skew-Symmetric Matrix

Orthogonal Matrix

A matrix A is said to be orthogonal if:

A^TA = I

A^T = A⁻¹

Example

A =

A^TA = I → Orthogonal Matrix

Important Properties

• Symmetric + Skew-Symmetric decomposition: A = (A + A^T)/2 + (A − A^T)/2
• Diagonal elements of skew-symmetric matrix = 0
• Determinant of orthogonal matrix = ±1
• Inverse of orthogonal matrix = Transpose

Solved Examples

Example 1

Check if matrix is symmetric:

Answer: Symmetric

Example 2

Check if matrix is skew-symmetric:

Answer: Skew-Symmetric

Example 3

Diagonal elements of skew-symmetric matrix?

Answer: 0

Example 4

If A is orthogonal, find A⁻¹

Answer: A^T

Shortcut Tricks

• Symmetric → A = A^T
• Skew → A = −A^T
• Orthogonal → A^TA = I
• Diagonal zero → skew-symmetric

Questions

Q1. If A is skew-symmetric, what is a₁₁?

Answer: 0

Q2. If A is orthogonal, determinant?

Answer: ±1

Exam Tips

• Memorize definitions clearly
• Focus on properties
• Avoid long calculations
• Practice matrix classification questions

🚀 Join Comprehensive Course

📢 GATE ESE PSU | ARJUNA Batch – Engineering Maths & General Aptitude

Prepare Engineering Mathematics & General Aptitude with Priyanka Sharma Ma’am (15+ Years Experience) at Studify+.

• ✔ Complete Engineering Mathematics coverage
• ✔ Complete General Aptitude preparation
• ✔ Live + Recorded Classes
• ✔ Concept clarity + practice questions
• ✔ 30 Years GATE PYQs (All Branches)
• ✔ Chapter & Subject Tests
• ✔ Designed for All GATE Branches

📚 Branches Covered:
ME | CE | EC | CS | EE | IN | MN | PE | XE | CH | PI | TF | AE | AG | GE & All

🔗 Explore Courses:
https://www.studifyplus.com/s/store

📱 Download App:
https://rb.gy/a3lwgu

📞 Contact:
WhatsApp / Call: 8200789441

✨ Build strong fundamentals and boost your GATE score!