Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are simil...
Let A = CD where C, D are n xn matrices, and is invertible. Prove that DC is similar to A. Hint: Use Theorem 6.13, and understand that you can choose P and P-inverse. Prove that if A is diagonalizable with n real eigenvalues 11, 12,..., An, then det(A) = 11. Ay n Prove that if A is an orthogonal matrix, then so are A and A'.
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
I will give a rate! please show work clearly! thanks! 12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A. 12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A.
Problem 3. Determine (with proof) whether each of the following statements is true or false. (a) For every m xn matrix A, det(AAT) = det(ATA) (b) Let A be an invertible n xn matrix, and suppose that B, C, and D are n x n matrices [det(A) |det(C) det (B) CA-1B. Then the 2 x 2 matrix is not invertible satisfying D (c) If A is an invertible n x n matrix such that A = A-1 then det(A) =...
oru 2 Let A and B be two n x n matrices. There exists a nonsingular matrix P such that PB = AP. Then which of the following is always true? a) A and B are not similar b) A and B have the same eigenvalues c) A does not have any characteristic polynomial d) B does not have any characteristic polynomial
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
(a) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any le R, we can write A = XI + (A - XI) (b) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn.n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M e Mn,n(R) such that M&V....
let a and b be n*n similar matrices, namely, B=S^-1 AS. show that the matrices a and b have the same characteristic polynomial, det(a-λI)=det(b-λI) and, consequently, the same eigenvalues.
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...