1. Prove the following statements: (a) If is an eigenvalue of a non-singular matrix A, then where A denotes the determinant of the matrix A. |A| is an eigenvalue of adj A, (b) If A and B are two invertible matrices, then AB and BA have same characteristic roots. (c) If is an eigenvalue of algebraic multiplicity r of A, then 0 is an eigenvalue of algebraic multiplicity r of the matrix A - In. 2. Find all the eigenvalues and the corresponding eigenvectors of the following matrices: 3 10 5 - -5 (a) 2 22) (0) (1) (c) 2 1 1 -2-3-4 (d) -1 2 -1 3 5 7 -1 2 3. Find two different 2 2 matrices A and B such that both have the same eigenvalues A = 2 = 2 and both have the same eigenvector corresponding to 2. 4. Let a+b=c+d. Show that () is an eigenvector of [a b] c d and find the eigenvalues. 5. Show that the matrix A = -i 3+2i -3+2i 0 2-i -3-4i -2-i 3 4i is skew-Hermitian. -2i