Let M be real vector space of order $2\times3$ matrices with the real entries. Let $T:M\longrightarrow M$ be defined by
$T\Bigg( \begin{pmatrix} x_{1} & x_{2} & x_{3} \\ x_{4} & x_{5} & x_{6}\end{pmatrix}\Bigg)$=$\begin{pmatrix}-x_{6}& x_{4}& x_{1}\\ x_{3}& x_{5}& x_{2}\end{pmatrix}$
how to calculate the determinant of T?
We can write $T = RS$, where $R$ is the map the multiplies the $(1, 1)$ entry by $-1$ and fixes all of the other entries and $S$ is the permutation matrix for the permutation $(16243)$ of six elements.
The matrix representation of $R$ is diagonal with entries $(-1, 1, \ldots 1)$, so $\det R = (-1)(1) \cdots (1) = -1$. The above permutation is even, so $\det S = 1$ and thus $$\det T = \det(RS) = \det R \det S = (-1)(1) = -1.$$