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A square matrix is called skew-symmetric if AT = -A. (a) (4 points) Explain why the...
please explain in full details. A square matrix A is skew-symmetric if A = -A (a)...
please explain in full
details.
A square matrix A is skew-symmetric if A = -A (a) If A is an n xn skew-symmetric matrix, with n odd, prove that A is singular, i.e. non-invertible (b) Find a skew-symmetric matrix that is invertible.
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of...
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)...
Question 3: (a) (4 points) Recall that the trace of a square matrix is the sum of all its entries...
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,...
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix 4. Show that an arbitrary square matrix A can be written as where Ai...
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
About discrete structure in CS class, a square matrix of dimension n is called a diagonal...
About discrete structure in CS class, a square matrix of dimension n is called a diagonal matrix if all cells except the left diagonal (cell positions (1, 1), (2, 2), (3, 3), …) contain 0. Suggest a quicker method (different from the standard method) to find the product of two diagonal matrices of size n.
Please answer the 25,26, and 27 25) A square matrix A = (a ) is called...
Please answer the 25,26, and 27
25) A square matrix A = (a ) is called diagonal if all its elements off the main diagonal are zero. That is, aij = 0 if j. (The matrix of Problem 24 is diagonal.) Show that a diagonal matrix is invertible if and only if each of its diagonal components is nonzero. 26.) Let a1i 0 0 0 a22 0 00ann be a diagonal matrix such that each of its diagonal components is...
(1 point) The trace of a square n x n matrix A = (aii) is the...
(1 point) The trace of a square n x n matrix A = (aii) is the sum ani + 022 + ... + ann of the entries on its main diagonal. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 1. Is Ha subspace of the vector space V? 1. Does H contain the zero vector of...
Explain all parts of question 1 and question 2 in detail 1. Consider the matrix In...
Explain all parts of question 1 and question 2 in detail
1. Consider the matrix In + Inn, which has every diagonal entry equal to 2 and every off-diagonal entry equal to 1. (a) Compute det(In + Inn) for each of n = 1,2,3. (b) For n = 4, we have 2 1 1 1 1 2 1 1 1 1 2 1 111 2 2 1 1 1 -1 1 0 0 -1 0 1 0 -1 0 0...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P su...
(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...
4. (5 points) Let A and B ben x n matrices. Prove that if A and...
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.
please explain in full
details.
A square matrix A is skew-symmetric if A = -A (a) If A is an n xn skew-symmetric matrix, with n odd, prove that A is singular, i.e. non-invertible (b) Find a skew-symmetric matrix that is invertible.
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)...
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,...
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
Please answer the 25,26, and 27
25) A square matrix A = (a ) is called diagonal if all its elements off the main diagonal are zero. That is, aij = 0 if j. (The matrix of Problem 24 is diagonal.) Show that a diagonal matrix is invertible if and only if each of its diagonal components is nonzero. 26.) Let a1i 0 0 0 a22 0 00ann be a diagonal matrix such that each of its diagonal components is...
(1 point) The trace of a square n x n matrix A = (aii) is the sum ani + 022 + ... + ann of the entries on its main diagonal. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 1. Is Ha subspace of the vector space V? 1. Does H contain the zero vector of...
Explain all parts of question 1 and question 2 in detail
1. Consider the matrix In + Inn, which has every diagonal entry equal to 2 and every off-diagonal entry equal to 1. (a) Compute det(In + Inn) for each of n = 1,2,3. (b) For n = 4, we have 2 1 1 1 1 2 1 1 1 1 2 1 111 2 2 1 1 1 -1 1 0 0 -1 0 1 0 -1 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...
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.
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