UPSC NDA Math Model Paper 10

Given, A is any square matrix and $A^{T}$ is transpose matrix $A$ Consider the statement "A $+A^{T}$ is always symmetric." We know that "a symmetric matrix is a square matrix that is equal to its transpose." i.e. $A = A^T .$ Consider, the matrix $A+A^{T} \\........(1)$ Taking transpose of the matrix $A+A^{T}$ $(A+A^{T})^{T}=(A)^{T}+(A^{T})^{T}$ $\Rightarrow (A+A^{T})^{T} = (A)^{T}+A$ $\Rightarrow (A+A^{T})^{T}= A + (A)^{T}$ $\Rightarrow A+A^{T}$ is symmetric. Hence, the statement "A $+A^{T}$ is always symmetric is true. Consider the statement "A - $A^{T}$ is always anti-symmetric" A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. i.e. $A^T = -A$ Consider, the matrix $A-A^{T} \\........(1)$ Taking transpose of the matrix $A-A^{T}$ $(A-A^{T})^{T} = (A)^{T} - (A^{T})^{T}$ $\Rightarrow (A-A^{T})^{T} = (A)^{T} - A$ $\Rightarrow (A-A^{T})^{T} = -(A-A^{T})$ $\Rightarrow A - A^{T}$ is symmetric. Hence, the statement "A - $A^{T}$ is always symmetric is true. Hence, the following statements in respect of a square matrix A and its transpose A :1. $A + A^T$ is always symmetric. 2. $A - A^T$ is always anti-symmetric. both are correct