Let A = [aij]n x n be a matrix. Then Match List-I with List-II
List-I
(A) AT = A
(B) AT = -A
(C) —A— = 0
(D) —A— ≠ 0
List-II
(I) A is a singular matrix
(II) A is a non-singular matrix
(III) A is a skew symmetric matrix
(IV) A is a symmetric matrix
Choose the correct answer from the options given below:
(A) - (IV), (B) - (III), (C) - (II), (D) - (I)
(A) - (IV), (B) - (III), (C) - (I), (D) - (II)
(A) - (I), (B) - (II), (C) - (III), (D) - (IV)
(A) - (I), (B) - (II), (C) - (IV), (D) - (III)
(A) - (IV), (B) - (III), (C) - (I), (D) - (II)
Step 1: Understanding the Concept:
This question tests the knowledge of basic definitions related to matrices, specifically symmetric, skew-symmetric, singular, and non-singular matrices.
Step 3: Detailed Explanation:
Let’s analyze each item in List-I and match it with the correct definition in List-II.
• (A) AT = A: This is the definition of a symmetric matrix. A matrix is symmetric if it is equal to its transpose. This matches with (IV).
• (B) AT = -A: This is the definition of a skew-symmetric matrix. A matrix is skewsymmetric if its transpose is equal to its negative. This matches with (III).
• (C) —A— = 0: The determinant of a matrix being zero is the condition for the matrix to be a singular matrix. This matches with (I).
• (D) —A— ≠ 0: The determinant of a matrix being non-zero is the condition for the matrix to be a non-singular matrix. Such matrices have an inverse. This matches with (II).
Step 4: Final Answer:
Combining the matches, we get: (A) → (IV) (B) → (III) (C) → (I) (D) → (II) This combination corresponds to option (2).