Finding the matrix of a permutation

Question

I am studying inverse permutations and I was given the following permutation sequence:

25413

I attempted to convert this permutation sequence into a permutation matrix:

I thought this was correct, but my book says that the permutation for 25413 is actually matrix A (which I thought was actually the inverse permutation matrix):

The permutation 25413 is represented by the 5x5 matrix A

I thought the permutation 25413 was represented by the matrix B. Can someone explain why 25413 is represented by matrix A?

Answer 1

$25413$ is presumably shorthand for \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 1 & 3 \end{pmatrix}

meaning that the position filled by element $1$ is subsequently filled by element $2$, etc., so your matrix needs to fulfill

$$A\cdot \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \\ a_5 \end{pmatrix} = \begin{pmatrix} a_2 \\ a_5 \\ a_4 \\ a_1 \\ a_3 \end{pmatrix} $$

Which is fairly obviously achieved by the row-swapped version of the identity matrix, and (perhaps less obviously) is undone by the column-swapped version.