Question

Match List-I with List-II 

List-I	Matrix Type	List-II	Property
A	Orthogonal matrix	I	Product of matrix and its transpose equals identity matrix
B	Skew-Hermitian matrix	IV	Diagonal elements are either zero or pure imaginary
C	Real skew symmetric matrix	II	Diagonal elements are zero
D	Hermitian matrix	III	Diagonal elements are real

 

• A. A-II, B-I, C-III, D-IV
• B. A-II, B-III, C-I, D-IV
• C. A-III, B-I, C-II, D-IV
• D. A-I, B-IV, C-II, D-III

Hint:

Orthogonal matrix satisfies \(AA^T=I\), while Hermitian matrix has real diagonal elements.

Answer 1

The Correct Option is D

Concept:
Different matrices are identified by their transpose, conjugate transpose and diagonal element properties.

Step 1: Orthogonal matrix. \[ AA^T=I \] So: \[ A\rightarrow I \]

Step 2: Skew-Hermitian matrix.
For skew-Hermitian matrix: \[ A^\dagger=-A \] Its diagonal elements are either zero or purely imaginary. \[ B\rightarrow IV \]

Step 3: Real skew symmetric matrix.
For real skew symmetric matrix: \[ A^T=-A \] Its diagonal elements are zero. \[ C\rightarrow II \]

Step 4: Hermitian matrix.
For Hermitian matrix: \[ A^\dagger=A \] Its diagonal elements are real. \[ D\rightarrow III \] \[ \therefore \text{Correct Answer is (D)} \]