Question

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| $\neq$ 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)

Hint:

Memorize the fundamental definitions of matrix types. The relationship between the determinant and singularity is a crucial concept in linear algebra. $|A| = 0 \Leftrightarrow$ Singular, $|A| \neq 0 \Leftrightarrow$ Non-singular.

Answer 1

The Correct Option is B

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) \( A^T = A \): This is the definition of a symmetric matrix. A matrix is symmetric if it is equal to its transpose. This matches with (IV).

(B) \( A^T = -A \): This is the definition of a skew-symmetric matrix. A matrix is skew-symmetric 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) \( |A| \neq 0 \): 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).