Exercise 1.3
"Sometimes it is useful to choose a key
such that the encryption operation is
identical to the decryption operation.
In the case of the Hill Cipher, we would
be looking for matrices K such that
K = K ^ {-1} (such a matrix is called
involutory). In fact, Hill recommended
the use of involutory matrices as keys in
his cipher. Determine the number of
involutory matrices (over Z_26) in the
case m = 2.
HINT Use the formula given in Theorem 1.3
and observe that det A = +-1 for an
involutory matrix over Z_26."
-Douglas R. Stinson-
See "Cryptography: Theory and Practice"
by Douglas R. Stinson page 41.

From Theorem 1.3 page 17 we have

A = | a[1][1] a[1][2] |
    | a[2][1] a[2][2] |

A ^ {-1} = (det A) ^ {-1} | a[2][2]   - a[1][2] |
                          | - a[2][1]   a[1][1] |

det A = a[1][1] * a[2][2] - a[1][2] * a[2][1] = +-1

a[1][1] = a[2][2]
a[1][2] = - a[1][2] = 0
a[2][1] = - a[2][1] = 0

So we have x = a[1][1] = a[2][2]

x * x = +-1 mod 26.

x = 1, 5, 21, 25. Thus we have four involutory
matrices of order m = 2 over Z_26.