A matrix with a non-zero determinant is invertable, and therefore usable in cryptographic purposes.
We encrypt with the following:
E = Ax
(where A, x are matrices)
And decrypt with the following:
D = A-1c
(where c is the encrypted text).
An encryption function is useless without a decryption function, so it is important that we know the matrix A has an inverse, which we can find out easily with the determinant, so we aren't wasting our time.
If we're working in a modulus, there are other properties we want from the determinant as well, for similar reasons (the encryption could produce the same ciphertext for two different inputs, that wouldn't be good).
Solving systems of equations is also a valid "real-world" use as solving systems of equations is necessary for economics, calculus (which does have real-world applications), and more.
