Consider the following statements :
I. If $n\times n$ $(n>1)$ matrix is symmetric, then its inverse is also a symmetric matrix.
II. If $n\times n$ $(n>1)$ matrix is singular, then its adjoint is also a singular matrix.
Which of the statements given above is/are correct ?

Let $A$ be an invertible symmetric matrix of order $n \times n$. Then $A^T = A$.

We know that $(A^{-1})^T = (A^T)^{-1}$.

Substituting $A^T = A$, we get $(A^{-1})^T = A^{-1}$.

Thus, the inverse of a symmetric matrix is also symmetric. The first statement is correct.

Let $A$ be a singular matrix of order $n \times n$. Then $|A| = 0$.

The determinant of the adjoint of $A$ is given by $|\text{adj}(A)| = |A|^{n-1}$.

Since $n > 1$ and $|A| = 0$, we have $|\text{adj}(A)| = 0^{n-1} = 0$.

Thus, the adjoint of a singular matrix is also a singular matrix. The second statement is correct.

Both statements are correct.

Answer: Both I and II