Please help! Prove thata) If \( S \) is invertible and \( k \) is a positive integer, show that \( S^k \) is invertible and \((S^k)^{-1} = (S^{-1})^k\).b) Let \( S \) be an invertible matrix and let \( A, B \) be matrices such that \( B = S^{-1}AS \). Show that \( B^k = S^{-1}A^kS \).

The objective of the question is to prove two properties of invertible matrices. The first part is to show that if a matrix S is invertible, then its kth power is also invertible and the inverse of the kth power of S is equal to the kth power of the inverse of S. The second part is to show that if B is a transformation of A by an invertible matrix S, then the kth power of B is equal to the transformation of the kth power of A by the same matrix S.Let's start with the first part. We know that the product of two invertible matrices is also invertible and the inverse of the product of two matrices is the product of the inverses in reverse order. That is, if S and T are invertible, then ST is invertible and (ST)^-1 = T^-1 S^-1. Now, consider S^k = S*S*...*S (k times). Since S is invertible, each factor in this product is invertible. Therefore, by the property mentioned above, S^k is invertible. Moreover, the inverse of S^k is the product of the inverses of the factors in reverse order, which is (S^-1)^k.Now, let's move to the second part. We are given that B = S^-1AS. We need to show that B^k = S^-1 A^k S. We can prove this by induction on k. For k=1, the statement is true because B^1 = S^-1 A^1 S. Now, assume that the statement is true for some positive integer k. That is, B^k = S^-1 A^k S. We need to show that the statement is true for k+1. We have B^(k+1) = B^k * B = (S^-1 A^k S) * (S^-1AS) = S^-1 A^k (SS^-1) A S = S^-1 A^k A S = S^-1 A^(k+1) S. Therefore, by the principle of mathematical induction, the statement is true for all positive integers k.In conclusion, we have proved that if a matrix S is invertible, then its kth power is also invertible and the inverse of the kth power of S is equal to the kth power of the inverse of S. Moreover, if B is a transformation of A by an invertible matrix S, then the kth power of B is equal to the transformation of the kth power of A by the same matrix S.