Minimum spanning trees given adjacency matrix of a graph

Consider the graph $$G$$ given by following adjacency matrix $$A=\begin{pmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 &1 &0 \\ 0 &1 &1 &0 & 1\\ 1&1 &0&1&0 \end{pmatrix}$$

Find the number of minimum spanning trees of the graph $$G$$

How do I find the number of minimum spanning trees? I can use Prim algorithm and find a minimum spanning tree but how do I know how many they are?

I draw the graph $$G$$ given its adjacency matrix:

You can use Kirchhoff's theorem. Let $D$ be the diagonal matrix whose entries are the degrees of the vertices of G. If you delete any row of $A-D$ (the Laplacian matrix) and its corresponding column, then the determinant of the resulting minor is the number of spanning trees for the graph.
We throw away the loop, since spanning trees don't have loops. Also, since all edges have weight 1, a "minimum spanning tree" is just a "spanning tree". A $n$-vertex spanning tree has $n-1$ edges; in this case that's $4$ edges. There are $\binom{7}{4}=35$ subgraphs with $4$ edges (ignoring the loop), which I drew using a script below.
First, remove $e_{12}$, and then compute MST for the other nodes ($k$). Then, you can multiply $k$ by 2, as there is a similar situation for removing $e_{15}$. Finally, suppose $e_{12}$ and $e_{15}$ exist in a MST. Now, compute MST between nodes $\{2, 3,4\}$ ($l$) and $\{3,4,5\}$ ($m$).
Hence, number of MST is $2k + l + m$ which comes from recursive existence of $e_{12}$, $e_{15}$ and both in the MST.