Suppose two n × n matrices A and B are similar.(a) (8 pts) Show that A and B have the same eigenvalues

Show that (8 pts) A2023 and B2023 are similar and (4 pts) have the same eigen-values.6 (10 pts) Let λ1 and λ2 be two distinct eigenvalues of an n × n matrix A. Let v1 and v2 betheir corresponding eigenvactors, respectively. Show that v1 and v2 are linearly independen

Let λk, k = 1, 2, ..., n, be the eigenvalues of an n × n invertible matrix A.(a) (2 pts) Show that λk 6 = 0, k = 1, 2, ..., n.(b) (8 pts) Show that the eigenvalues of A−1 are 1λk, k = 1, 2, ..., n

waht are similar matrices explain it with an example

Suppose that A and B are n×n matrices. Check the true statements below: A. If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.B. det(AT)=(−1)det(A)C. If two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determinant of the original matrix.D. The determinant of A is the product of the diagonal entries in A

rue or false:a) Every linear operator in an n-dimensional vector space has n distinct eigen-values;b) If a matrix has one eigenvector, it has infinitely many eigenvectors;c) There exists a square real matrix with no real eigenvalues;d) There exists a square matrix with no (complex) eigenvectors;e) Similar matrices always have the same eigenvalues;f) Similar matrices always have the same eigenvectors;g) A non-zero sum of two eigenvectors of a matrix A is always an eigenvector;h) A non-zero sum of two eigenvectors of a matrix A corresponding to thesame eigenvalue λ is always an eigenvector