Which of the following statement(s) is/are true with respect to eigenvalues andeigenvectors of a matrix?(A) The sum of the eigenvalues of a matrix equals the sum of the elements of theprincipal diagonal.(B) If is an eigenvalue of a matrix A, then1 is always an eigenvalue of itstranspose (AT).(C) If is an eigenvalue of an orthogonal matrix A, then1 is also an eigenvalue ofA.(D) If a matrix has n distinct eigenvalues, it also has n independent eigenvectors.Q.30 For studying wing vibrations, a wing of mass M and finite dimensions has beenidealized by assuming it to be supported using a linear spring of equivalentstiffness k and a torsional spring of equivalent stiffnessk as shown in the figure.The centre of gravity (CG) of the wing idealized as an airfoil is markedin the figure. The number of degree(s) of freedom for this idealized wing vibrationmodel is _________. (Answer in integer)

To determine which of the statements are true with respect to eigenvalues and eigenvectors of a matrix, let's analyze each statement one by one:

(A) The sum of the eigenvalues of a matrix equals the sum of the elements of the principal diagonal. This statement is not true in general. The sum of the eigenvalues of a matrix is equal to the trace of the matrix, which is the sum of the elements on the principal diagonal. However, the sum of the eigenvalues may not necessarily be equal to the sum of all elements on the principal diagonal.

(B) If λ is an eigenvalue of a matrix A, then 1/λ is always an eigenvalue of its transpose (AT). This statement is true. If λ is an eigenvalue of matrix A, then there exists a non-zero eigenvector x such that Ax = λx. Taking the transpose of both sides, we have (AT)x = (1/λ)x, which shows that 1/λ is an eigenvalue of AT.

(C) If λ is an eigenvalue of an orthogonal matrix A, then 1/λ is also an eigenvalue of A. This statement is not true. For an orthogonal matrix, the eigenvalues can only be +1 or -1. Therefore, if λ is an eigenvalue of an orthogonal matrix A, then 1/λ will not be an eigenvalue of A.

(D) If a matrix has n distinct eigenvalues, it also has n independent eigenvectors. This statement is true. If a matrix has n distinct eigenvalues, then each eigenvalue corresponds to a unique eigenvector. Since the eigenvalues are distinct, the eigenvectors corresponding to different eigenvalues will be linearly independent.

Moving on to the second part of the question, the number of degrees of freedom for the idealized wing vibration model can be determined by counting the number of independent variables that can describe the motion of the system. In this case, we have two independent variables: the displacement of the wing in the vertical direction (related to the linear spring) and the angular displacement of the wing (related to the torsional spring). Therefore, the number of degrees of freedom for this idealized wing vibration model is 2.