Let A = $[a_{ij}]_{n \times n}$ be a matrix. Then

Match List-I with List-II

List-I	List-II
(A) $A^T = A$	(I) A is a singular matrix
(B) $A^T = -A$	(II) A is a non-singular matrix
(C) $\vert A\vert = 0$	(III) A is a skew symmetric matrix
(D) $\vert A\vert \neq 0$	(IV) A is a symmetric matrix

Choose the correct answer from the options given below:

Let's match the matrix properties with their correct definitions.

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First, let's understand what each property in List-I means:

$(A)$ $A^T = A$ means when you transpose the matrix (flip across diagonal), it stays the same

$(B)$ $A^T = -A$ means when you transpose the matrix, every element changes sign

$(C)$ $|A| = 0$ means the determinant of the matrix equals zero

$(D)$ $|A| \neq 0$ means the determinant of the matrix is not zero

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Now let's recall the definitions in List-II:

$(I)$ Singular matrix: A matrix with determinant $= 0$

$(II)$ Non-singular matrix: A matrix with determinant $\neq 0$

$(III)$ Skew symmetric matrix: A matrix where $A^T = -A$

$(IV)$ Symmetric matrix: A matrix where $A^T = A$

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Matching them directly:

$(A)$ $A^T = A$ matches with $(IV)$ Symmetric matrix

$(B)$ $A^T = -A$ matches with $(III)$ Skew symmetric matrix

$(C)$ $|A| = 0$ matches with $(I)$ Singular matrix

$(D)$ $|A| \neq 0$ matches with $(II)$ Non-singular matrix

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Answer: $A \rightarrow IV$, $B \rightarrow III$, $C \rightarrow I$, $D \rightarrow II$