3. (5 points) Find a basis for all "skew-symmetric matrices. For your reference, if AT =...
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...
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)...
4. (5 points) Let A and B ben x n matrices. Prove that if A and B are skew symmetric, then A - B is skew symmetric. Recall C = [cj] is skew symmetric iff Cij =-Cji.
Question 3: (a) (4 points) Recall that the trace of a square matrix is the sum of all its entries from the main diagonal. Show that the trace is linear, in the sense that, trace(aX + βΥ) trace(X) + β trace(Y). Let V be the space of all m × n matrices. A function <..) : V × V → R is defined as (A, B) trace(ABT), A, B E V. (a) (4 points) Using the properties of the trace,...
Help me plz to solve questions a and b 9. (10pts) Answer only four parts by True/False and provide justifica- tions] Given A, B and C three n × n matrices: (a) If C'is a nonsingular skew-symmetric matrix, then its inverse is also skew symmetric b) If rank(A) and AB- AC then B- C c) Let S-V, V2, Vs) be a lnearly independent set of vectors in a vector space V and T V2, V2+Vs, ViVs); then T is linearly...
2. Find a basis for the set of all 2 x 2 symmetric ma trices. Yo s really is a basis. Use your answer to find the dimension of the set of all symmetric 2 x 2 matrices.. u should give convincing evidence that your
7. (10) Find all n xn orthogonal, symmetric, and positive definite real matrix (matrices). Explain your answer.
QUESTION 6 a) Prove the product of 2 2 x 2 symmetric matrices A and B is a symmetric matrix if and only if AB=BA. b) Prove the product of 2 nx n symmetric matrices A and B is a symmetric matrix if and only if AB=BA.
linear algebra Find all n x n orthogonal, symmetric, and positive definite real matrix (matrices). Explain answer
Show each of the 3 following matrices is symmetric and idempotent ( J is a matrix with all 1 s) For the next few problems, let X = (X1X2), ßT = (β.β;), H the lat matrix for X, and Hi the hat matrix for X i. (I 1/nJ) ii. (I H)