If $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$, show that $A-A^{T}$ is a skewsymmetric matrix.
Given : $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$
$A^{T}=\left[\begin{array}{cc}3 & 1 \\ -4 & -1\end{array}\right]$
Now,
$A-A^{T}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]-\left[\begin{array}{cc}3 & 1 \\ -4 & -1\end{array}\right]$
$\Rightarrow A-A^{T}=\left[\begin{array}{ll}3-3 & -4-1 \\ 1+4 & -1+1\end{array}\right]$
$\Rightarrow A-A^{T}=\left[\begin{array}{cc}0 & -5 \\ 5 & 0\end{array}\right]$ ....(1)
$\left(A-A^{T}\right)^{T}=\left[\begin{array}{cc}0 & -5 \\ 5 & 0\end{array}\right]^{T}$
$\Rightarrow\left(A-A^{T}\right)^{T}=\left[\begin{array}{cc}0 & 5 \\ -5 & 0\end{array}\right]$
$\Rightarrow\left(A-A^{T}\right)^{T}=-\left[\begin{array}{cc}0 & -5 \\ 5 & 0\end{array}\right]$
$\Rightarrow\left(A-A^{T}\right)^{T}=-\left(A-A^{T}\right)$
$[$ From eq. $(1)]$
Thus, $\left(A-A^{T}\right)$ is a skew - symmetric matrix.
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