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 = 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 A? = 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.
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.
Let C be square matrix. i) Check if S = C + CT is symmetric ii) Check if N + C - CTis skew symmetric iii) Prove that every square matrix can be written as a sum of skew symmetric matrix and symmetric matrix
[3] 7. Let A be a square matrix such that A# I and A+ -I with eigenvalue X. Prove that if AP = I(I is the identity matrix), then = 1 or = -1.
(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...
(3) 7. Let A be a square matrix such that A# I and A+ - with eigenvalue A. Prove that if AP = (is the identity matrix), then X = 1 or X = -1.
Let A be a square matrix. Prove that if A2 = A, then I - 2A is the inverse of I - 2A.