Trace, Transpose & Conjugate of Matrix – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

Trace, Transpose, and Conjugate of Matrix are fundamental topics in Engineering Mathematics and are frequently asked in GATE, ESE, AE/JE exams. These concepts form the base for advanced topics in Linear Algebra.

Basic Concepts

A matrix is a rectangular arrangement of numbers in rows and columns.

Transpose of a Matrix

The transpose of a matrix is obtained by interchanging its rows and columns.

If A = [a_ij], then A^T = [a_ji]

Example

A = [ 1 2
     3 4 ]

A^T = [ 1 3
     2 4 ]

Trace of a Matrix

The trace of a square matrix is the sum of its principal diagonal elements.

Trace(A) = a₁₁ + a₂₂ + a₃₃ + ...

Example

A = [ 2 1
     3 4 ]

Trace(A) = 2 + 4 = 6

Conjugate of a Matrix

The conjugate of a matrix is obtained by taking the complex conjugate of each element.

If z = a + ib → Conjugate(z) = a − ib

Example

A = [ 2 + i , 3 − i ]

Conjugate(A) = [ 2 − i , 3 + i ]

Important Properties

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (AB)^T = B^TA^T
• Trace(A + B) = Trace(A) + Trace(B)
• Trace(AB) = Trace(BA)

Solved Examples

Example 1

Find transpose of A = [1 2; 3 4]

A^T = [1 3; 2 4]

Example 2

Find trace of matrix [5 1; 2 6]

Trace = 5 + 6 = 11

Example 3

Find conjugate of (4 − 3i)

= 4 + 3i

Example 4

If Trace(A) = 12 and Trace(B) = 8, find Trace(A + B)

Trace(A + B) = 12 + 8 = 20

Shortcut Tricks

• Transpose → swap rows & columns
• Trace → sum only diagonal elements
• Conjugate → change sign of imaginary part
• Use properties to save time

Questions

Q1. Find trace of identity matrix of order 4.

= 1 + 1 + 1 + 1 = 4

Q2. If A = A^T, identify matrix type.

Symmetric Matrix

Exam Tips

• Focus on fundamental properties
• Avoid lengthy calculations
• Practice matrix-based questions
• Revise formulas regularly

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