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 ?
C. Both I and II
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 correct. Both statements are correct. Answer: Both I and II
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