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A possibility to give a sense of the matrix $M$ is sage is as follows. Consider the field $F=\Bbb F_{13}$ with $13$ elements. The number $7$, seen as an element of this field, is not a square.

sage: F = GF(13)
sage: F(7).is_square()
False

So in order to have the element $5+\sqrt 7$, we have to construct the field extension $L=\Bbb F_{169}$, pick some element $A\in L$, with $A^2=7$, and consider the matrix $M$ with entries $5+A$, $0$, $0$, $5-A$. We have:

F = GF(13)
R.<x> = PolynomialRing(F)
L = GF(13^2)
A = [A for A in L if A^2 == 7][0]
M = matrix(L, 2, 2, [5 + A, 0, 0, 5 - A])

print(f"M has multiplicative order {M.multiplicative_order()}")

And we obtain:

M has multiplicative order 56
sage: M^56
[1 0]
[0 1]
sage:

Also:

sage: (5 + A).multiplicative_order()
56
sage: (5 - A).multiplicative_order()
56

So $M$ has order $56$ seen as an element of $GL(2, L)$, in particular, its order in $PGL(2,L)$ is a divisor of $56$. Let us compute the powers of $M$ for all divisors of $56$, this is the same as doing so only for the conjugated numbers $5\pm A$ in $L$:

sage: [(d, (5-A)^d) for d in 56.divisors()]
[(1, 10*z2),
 (2, 9*z2 + 8),
 (4, 4*z2 + 6),
 (7, 5*z2 + 4),
 (8, 12*z2 + 4),
 (14, 5),
 (28, 12),
 (56, 1)]
sage:

It is clear now, that $M^{14}$ acts as a scalar, so it is an element of the center $Z$ of $GL(2,L)$, so $M$ seen as an element of $PGL(2,L)=GL(2,L)/Z$ is the identity. (And for no divisor of $14$ "the same" happens.)

sage: M^14
[5 0]
[0 5]

Note: It is strange, but the code

F = GF(13)
R.<x> = PolynomialRing(F)
L.<a> = GF(13^2, modulus=x^2-7)
M = matrix(L, 2, 2, [5 + a, 0, 0, 5 - a])

delivers a false multiplicative order $168$ for $M$, however the right order $56$ for $(5\pm a)$.

sage: M
[   a + 5        0]
[       0 12*a + 5]
sage: M.multiplicative_order()
168
sage: M^56
[9 0]
[0 9]
sage: (5 + a)^56
1
sage: (5 + a).multiplicative_order()
56
sage: M^2
[11*a + 10         0]
[        0  4*a + 10]
sage: (5 + a)^2
10*a + 6

This is the reason for the rather cumbersome introduction of $A$ and $L$ in the "good part of the answer".