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

Step-by-step solution

Let A be an invertible symmetric matrix of order n 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 n . Then |A| = 0 . The determinant of the adjoint of A is given by | adj (A)| = |A|^ n-1 . Since n > 1 and |A| = 0 , we have | adj (A)| = 0^ n-1 = 0 . Thus, the adjoint of a singular matrix is also a singular matrix. The second statement is c