Consider the following statements in respect of a square matrix A and its transpose AT : 1.A+AT is always symmetric. 2.A−AT is always anti-symmetric. Which of the statements given above is / are correct?
Given, A is any square matrix and AT is transpose matrix A Consider the statement "A +AT is always symmetric." We know that "a symmetric matrix is a square matrix that is equal to its transpose." i.e. A=AT. Consider, the matrix A+AT........(1) Taking transpose of the matrix A+AT (A+AT)T=(A)T+(AT)T ⇒(A+AT)T=(A)T+A ⇒(A+AT)T=A+(A)T ⇒A+AT is symmetric. Hence, the statement "A +AT is always symmetric is true. Consider the statement "A - AT is always anti-symmetric" A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. i.e. AT=−A Consider, the matrix A−AT........(1) Taking transpose of the matrix A−AT (A−AT)T=(A)T−(AT)T ⇒(A−AT)T=(A)T−A ⇒(A−AT)T=−(A−AT) ⇒A−AT is symmetric. Hence, the statement "A - AT is always symmetric is true. Hence, the following statements in respect of a square matrix A and its transpose A :1. A+AT is always symmetric. 2. A−AT is always anti-symmetric. both are correct