Determinants Exist

Give the permutation expansion of a general $2\!\times\!2$ matrix and its transpose.

This problem appears also in the prior subsection.

• Find the inverse of each $2$-permutation.

• Find the inverse of each $3$-permutation.

• Find the signum of each $2$-permutation.

• Find the signum of each $3$-permutation.

Find the only nonzero term in the permutation expansion of this matrix.

Compute that determinant by finding the signum of the associated permutation.

What is the signum of the $n$-permutation $\phi=\langle n,n-1,\dots,2,1\rangle$?

Prove these.

• Every permutation has an inverse.

• $\operatorname{sgn}(\phi^{-1})=\operatorname{sgn}(\phi)$

• Every permutation is the inverse of another.

Prove that the matrix of the permutation inverse is the transpose of the matrix of the permutation $P_{\phi^{-1}}={{P_{\phi}}^{\mathsf{T}}}$, for any permutation $\phi$.

Show that a permutation matrix with $m$ inversions can be row swapped to the identity in $m$ steps. Contrast this with Corollary 4.5.

For any permutation $\phi$ let $g(\phi)$ be the integer defined in this way.

(This is the product, over all indices $i$ and $j$ with $i<j$, of terms of the given form.)

• Compute the value of $g$ on all $2$-permutations.

• Compute the value of $g$ on all $3$-permutations.

• Prove that $g(\phi)$ is not $0$.

• Prove this.

Many authors give this formula as the definition of the signum function.