Eigenvalues of Similar Matrices

Solution: Given matrix is
a)
Note that, if A is any 3 by 3 matrix such that , then by using Laplace-
expansion by column 3, the determinant of the matrix A is
So for the given matrix,
Hence, the determinant of given matrix is 0.
b (i): we know that a matrix is non-singular if its determinant is non zero. Since the
determinant of the given matrix is 0. Hence, matrix A is not singular, that is matrix is
singular.
b (ii): The trace of A10 is the sum of diagonal elements of matrix A10.
c): Given matrix is
(i): The determinant of matrix B is
(ii): Cofactors
(iii): the cofactor matrix is
So, the adjoint of the matrix B is the transpose of matrix P. that is
(iv): The inverse of matrix B is

Solution 3:
Given matrix is
(a):
To find eigenvalues of A, solve this implies that
This implies that the eigenvalues are
(b): Since, the largest eigenvalue is . So solve we get
Now, apply gauss elimination method to find value of eigenvector u.
We get
Suppose that
This implies that
Hence, for , eigenvector corresponding to the eigenvalue is
Therefore, the eigenvector of unit length corresponding to the largest eigenvalue of A is
(c): Since, the 2nd largest eigenvalue is . So solve we get
Now, apply gauss elimination method to find value of eigenvector u.
We get
Suppose that
This implies that