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 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 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.
Let A be a square matrix. Prove that if A2 = A, then I - 2A is the inverse of I - 2A.
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
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.
(1 point) A matrix A is said to be similar to a matrix B if there is an invertible matrix P such that B = PAP 1 Let A1, A2, and A3 be 3 x 3 matrices Prove that if A1 is similar to A2 and A2 is similar to A3, then A similar to A. Proof: Since A1 is similar to A2, for some invertible matrix P for some invertible matrix Q Since A2 is similar to A3 for...
(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...
A matrix A is said to be idempotent if A2 - A. Show that each of the following is idempotent. 1 nJ H-X(XX)- I-H I-J