Permutations of elements of a matrix

Question

I'm a bit confused about the second part of the question I'm working on. The question is as follows

Let A be the $4 \times 4$ matrix $$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}$$

(a) Write out all of the permutations $\sigma \in S_4$ with $\sigma(1)=4$ (there are six of them)

(b) Using the general formula for det($A$), write out all the terms corresponding to the permutations you found in part (a)

From my notes I have

To describe $\sigma$ in $S_n$ write $1,2,...,n$ $$\begin{pmatrix} 1 & 2 & 3 & \cdots & 5 \\ \sigma(1) & \sigma(2) & \sigma(3) & \cdots & \sigma(n) \end{pmatrix}$$

So the six permutations I end up with are

$$\begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 2 & 1 & 3 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 3 & 1 & 2 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 3 & 2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 2 & 3 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 1 & 3 & 2 \end{pmatrix}$$

Now, I am not sure about part (b). I have 6 different permutations of 4 numbers. What terms from the general formula for det($A$) correspond to the permutations I found?

Answer 1

I believe the following is meant:  

The general formula (complete expansion) of the determinant is:

$$\det A = \sum_{\textrm{permutations p}} \left( \textrm{sign p} \right) a_{1, p_1} a_{2, p_2} \textrm{...} a_{n, p_n}.$$

In the case of your first permutation, the term would thus be:  

$$\left( -1 \right) a_{1, 4} a_{2, 1} a_{3,2} a_{4, 3},$$

with the others being similarly translated.