Let P be an orthogonal matrix.
(a) Prove that detP = 1 or detP = −1.
(b)If detP = −1, show that I+P has no inverse. Hint:
PT(I+P)=(I+P)T. (which T means transpose)
Let P be an orthogonal matrix. (a) Prove that detP = 1 or detP = −1....
2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is orthogonal, that is PT P I, 2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is orthogonal, that is PT P I,
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
Help on Questions 1-3 Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an orthogonal matrix. Show also that the vector Show that the matrix A is an eigenvector for the matrix A and determine the corresponding eigenvalue 27. Prove that the determinant of the matrix 2 Y3 -I is 2, where (y)(y2()(ys)2. Prove also that the inverse of the matrix G is G(G-I)T İs an...
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and let A be an n x n skew-symmetric matrix. Simplify A(A^2(BA)^-1)^T b. Let A be a square matrix such that A^3 = 0. A is then called a nilpotent matrix. Define another matrix B by the expression B = I - A; Show that B is invertible and that its inverse is I + A + A^2 c. Let B = (-2,0,0 ; 0,0,0...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
1 Consider the matrix P= 1 o 1 Which one of the following statements about P is INCORRECT? (A) Determinant of P is equal to 1. (B) Pis orthogonal. (C) Inverse of P is equal to its transpose. (D) All Eigen values of P are real numbers
[12] QUESTION 4 (a) Let A be an m × m symmetric matrix and P be an orthogonal matrix such the PAP-D,where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that PA P is also a diagonal matrix. (b) Let A be an m × n matrix of rank m such that A = BC where B and C each has rank m. Show that (BC) CB. 16 STA4801/101/0/2019 (c) For the matrix...
Let A be an m x n matrix and let B be an n x p matrix. (a) Prove that Col(AB) SColA) (b) Use part (a) to prove that the rank of AB is at most the rank of A (c) Use transpose matrices to prove that the rank of AB is also at most the rank of B.
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'.