Question

Let $A$ be a square matrix. Consider
1 $)A+A^T$ 2 $\left)A{A}^{T}3\right)A^{T}A$ 4 $)A^T+A$
5 $)A-A^T$ 6 $)A^T-A$, Then

• A. all are symmetric matrices
• B. $\left(2\right),\left(4\right),\left(6\right)$ are symmetric matrices
• C. $\left(1\right),\left(2\right),\left(3\right),\left(4\right)$ are symmetric matrices &
  $\left(5\right),\left(6\right)$ are skew symmetric matrices
• D. $5,6$ are symmetric

Answer 1

The correct option is C $\left(1\right),\left(2\right),\left(3\right),\left(4\right)$ are symmetric matrices &

$\left(5\right),\left(6\right)$ are skew symmetric matrices
1)Consider, $(A+A^T{)}^T=A^T+(A^T{)}^T$
$=A^T+A$
$=A+A^T$
$\Rightarrow (A+A^T{)}^T=A+A^T$
Hence, $A+A^T$ is symmetric
2)$(A{A}^T{)}^T=(A^T{)}^T{A}^T$
$=A{A}^T$
$\Rightarrow (A{A}^T{)}^T=A{A}^T$
Hence, $A{A}^T$ is symmetric.
3) $(A^{T}A{)}^T=A^T(A^T{)}^T$
$=A^{T}A$
$\Rightarrow (A^{T}A{)}^T=A^{T}A$
Hence, $A^{T}A$ is symmetric
4)$(A^T+A{)}^T=(A^T{)}^T+A^T$
$=A+A^T$
$=A^T+A$
$\Rightarrow (A^T+A{)}^T=A^T+A$
Hence, $A^T+A$ is symmetric
5) $(A-A^T{)}^T=A^T-(A^T{)}^T$
$=A^T-A$
$=-(A-A^T)$
$\Rightarrow (A-A^T{)}^T=-(A-A^T)$
Hence, $A-A^T$ is skew-symmetric
6) $(A^T-A{)}^T=(A^T{)}^T-A^T$
$=A-A^T$
$=-(A^T-A)$
$\Rightarrow (A^T-A{)}^T=-(A^T-A)$
Hence, $A^T-A$ is skew-symmetric