Example Find the determinant of the following matrix．

$$\left \vert \begin{array}{ccccc} 0&0&0&1&0\\ 0&1&0&0&0\\ 0&0&0&0&1\\ 1&0&0&0&0\\ 0&0&1&0&0 \end{array}\right\vert$$

Answer

$${} \left\vert\begin{array}{rrrrr} 0&0&0&1&0\\ 0&1&0&0&0\\ 0&0&0&0&1\\ 1&0&0&0&0\\ 0&0&1&0&0 \end{array}\right\vert$$

$$= {\rm sgn}(4,2,5,1,3)1\cdot 1\cdot 1\cdot 1 \cdot 1 = - sgn(1,2,5,4,3) = sgn(1,2,3,4,5) = +1$$

Note that $(4,2,5,1,3)$ represents the permutation order. The first number 4 means that the first nonzero digit from the top row appears in the 4th column. Similarly, the number 2 means that the second nonzero digit from the top appears in the 2nd column and so forth.

We then explain the symbol sgn. the symbol sgn gives rise to either $+$ or $-$ depending on the number of transposition. If the number of transposition is even, then the sgn is $+$ and if the number of transposition is odd, then the sgn is $-$. We note that the number of transposition may not the same, but the even or odd is unique.