: (a) For what values of a, b, and c is the following matrix symmetric? [-3...
A square matrix is called skew-symmetric if AT = -A. (a) (4 points) Explain why the main diagonal of a skew-symmetric matrix consists entirely of zeros. (b) (2 points) Provide examples of a 2 x 2 skew-symmetric matrix and a 3 x 3 skew-symmetric matrix. (6 points) Prove that if A and B are both n x n skew-symmetric matrices and c is a nonzero scalar, then A + B and cA are both skew-symmetric as well. (4 points) Find...
Exercise 10. Write A = 1-3 6 01 as the sum of a symmetric matrix B and a skew-symmetric matrix C. SVImImetric matrIX C
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l A^T=-A}. (a) Show that W is a subspace of M2x2(R) (b) Find a basis for W and determine dim(W). (c) Suppose T: M2x2(R) is a linear transformation given by T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You do not need to verify that T is linear. 3. (17 points)...
Problem # 1: (70 points) Solve the following problems (a) and (b) using Laplace Transform: a) (7 points) y(0)-y'(0)-0 y"(0)-1 b) (dX/d't) + 3 (dy/dt) + 3y-0 (7 points) (d'x/d't) +3y-te' x(0) = 0 x'(0) = 2 y(0) = 0 c) An nxn matrix A is said to be skew-symmetric if AT--A. If A is a 5x5 skew-symmetric matrix, show that 9detA)-0 (4 Points) d) Suppose A is a 5x5 matrix for which (detA) =-7, what is the value of...
2 invertible? C For which values of c is the matrix 8 O c 4 c =-4 Both of the above, i.e., c +4 Neither of the above, i.e., c +4. Suppose that the following row operations: interchange rows 1 and 3 multiply row 3 by 1/2 add -3 times row 1 to row 2 2 1 7 in this order, transform a matrix A into B = | 0 4-5 L0 0 3 What is the determinant of A?...
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...
Problem 4: Suppose A = (ai)nxn is a symmetric matrix (i.e. the transpose of A agrees with itself) and a11 +0. After we use a11 to eliminate a21, ... , Anl, we obtain a matrix of the following form: (n-1)-matrix. Here c is an (n-1)-dimensional column vector and ct is its transpose, while B is an (n-1) Prove that B is also symmetric.
[3 marks] Consider the following statements (1) If AT A is a symmetric matrix, then A must be a square matrix. (ii) If A is nx n then A'(A ) - 1. (iii) If A is an nxn matrix, then tr(CA) - ctr(A). Determine which of the above statements are True (1) or False (2) So, for example, if you think that the answers, in the above order, are True False False, then you would enter "1.2.2' into the answer...
please help me answer this question Lecky Nunber Memnon Cless M 2. Which matrix is not an elementary matrix? 100 100 100 1 1 10 (D). (C). (B). 01 4 001 (A). 0 1 0 00 1 0 1 1 001 3. Which matrix is invertible? 2 3 100 -7 0 3 [1 2 3 (D). 1 2 3 6 4 (C). 0 0 3 3 01 (A). 3 5 9 6 8 18 (B). 004 2 04 4-5a, a-5a...