Homework Answers
If you have doubt in any step please comment down i will try to explain that step and if you were able to understand the explanation please upvote
The converse of above theoren is not true as null matrix is
symmetric matrix but is neither orthogonal nor involutary and the
proof of if matrix is orthogonal and involutary both then it is
symmetric is done below
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7. Matrix A is said to be involutory if A? = 1. Prove that a square...
7. Matrix A is said to be involutory if A² = 1. Prove that a square...
7. Matrix A is said to be involutory if A² = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
7. Matrix A is said to be involutory if A2 = 1. Prove that a square...
7. Matrix A is said to be involutory if A2 = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
7. Matrix A is said to be involutory if AP = 1. Prove that a square...
7. Matrix A is said to be involutory if AP = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
7. Matrix A is said to be involutory if A² = I. Prove that a square...
7. Matrix A is said to be involutory if A² = I. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
17 (a) Prove that a permutation π in the Permutation Cipher is an involutory key if...
17 (a) Prove that a permutation π in the Permutation Cipher is an involutory key if and only if π(i) = j implies π(j) = i, for all i, j E {1, . . . , m} (b) Determine the number of involutory keys in the Permutation Cipher for m = 2,3, 4, 5 and 6.
(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...
Help on Questions 1-3 Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x...
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 +...
Let A be an invertible matrix, prove that A is symmetric if and only if A-1...
Let A be an invertible matrix, prove that A is symmetric if and only if A-1 is symmetric.
A square matrix is called skew-symmetric if AT = -A. (a) (4 points) Explain why the...
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...
29. A matrix B is said to be a square root of a matrix A if...
29. A matrix B is said to be a square root of a matrix A if BB A (a) Find two square roots of A = (b) How many different square roots can you find of - (c) Do you think that every 2 x 2 matrix has at least one square root? Explain your reasoning
7. Matrix A is said to be involutory if A² = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
7. Matrix A is said to be involutory if A2 = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
7. Matrix A is said to be involutory if AP = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
7. Matrix A is said to be involutory if A² = I. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
(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...
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 +...
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...
29. A matrix B is said to be a square root of a matrix A if BB A (a) Find two square roots of A = (b) How many different square roots can you find of - (c) Do you think that every 2 x 2 matrix has at least one square root? Explain your reasoning
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