Sarrus-determinant/4x4/Properties/Exercise

The Sarrusminant of a ${}n\times n$-matrix is computed by repeating the first ${}n-1$ columns of the matrix in the same order behind the matrix, and then by adding up the ${}n$ products of the diagonals and subtracting the ${}n$ products of the antidiagonals. We restrict to the case ${}n=4$. That is, for a matrix

${}M={\begin{pmatrix}a&b&c&d\\e&f&g&h\\l&m&n&m\\q&r&s&t\end{pmatrix}}\,,$

we consider

${\begin{pmatrix}a&b&c&d&a&b&c\\e&f&g&h&e&f&g\\l&m&n&m&l&m&n\\q&r&s&t&q&r&s\end{pmatrix}}$

and the Sarrusminant is

${}\operatorname {Sar} (M)=afnt+bgpq+chlr+dems-qmgd-rnha-speb-tlfc\,.$

1. Show that the mapping

   $\Psi \colon \operatorname {Mat} _{4}(K)\longrightarrow K,M\longmapsto \operatorname {Sar} (M),$

   is multilinear (in the rows of the matrix).

2. Show that, for ${}4\times 4$-matrices that contain a zero-row, the Sarrusminant is ${}0$.
3. Show that, for ${}4\times 4$-matrices that contain a zero-column, the Sarrusminant is ${}0$.
4. Show that, for an upper triangular matrix, the Sarrusminant is the product of the diagonal elements.
5. Show that the Sarrusminant is not alternating.
6. Give an example for an invertible matrix, where the Sarrusminant equals ${}0$.
7. Give an example for a not-invertible matrix, where the Sarrusminant equals ${}1$.

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