Matrix Properties Explained

This question tests the understanding of fundamental matrix properties related to the transpose of a matrix ($A^T$) and its determinant ($|A|$).

Understanding Matrix Types

• Symmetric Matrix: A square matrix $A$ is symmetric if its transpose is equal to the matrix itself. Mathematically, this is represented as $A^T = A$. The elements across the main diagonal are mirrored.
• Skew-Symmetric Matrix: A square matrix $A$ is skew-symmetric if its transpose is equal to the negative of the matrix. Mathematically, this is represented as $A^T = -A$. For a skew-symmetric matrix, the diagonal elements must be zero.
• Singular Matrix: A square matrix $A$ is singular if its determinant is zero. Mathematically, this is $|A| = 0$. Singular matrices do not have an inverse.
• Non-singular Matrix: A square matrix $A$ is non-singular if its determinant is not zero. Mathematically, this is $|A| \neq 0$. Non-singular matrices have a unique inverse.

Matching Matrix Conditions to Definitions

We need to match the conditions given in List-I with their corresponding definitions or properties in List-II.

• (A) $A^T = A$: This condition defines a symmetric matrix. Therefore, (A) matches with (IV).
• (B) $A^T = -A$: This condition defines a skew-symmetric matrix. Therefore, (B) matches with (III).
• (C) $|A| = 0$: This condition signifies that the matrix $A$ has a determinant of zero, which means it is a singular matrix. Therefore, (C) matches with (I).
• (D) $|A| \neq 0$: This condition signifies that the matrix $A$ has a non-zero determinant, which means it is a non-singular matrix. Therefore, (D) matches with (II).

Identifying the Correct Option

Based on the matching derived above, the correct pairings are:

• (A) - (IV)
• (B) - (III)
• (C) - (I)
• (D) - (II)

This set of matches corresponds to Option 2.