For a matrix A to be symmetric, A = A^T, which is possible if b = c.
Hence we get |A| = a² – b²
We must have a² – b² = kp
⇒ (a + b)(a – b) = kp
⇒ either a – b or a + b is a multiple of p
a – b will be divisible by p only when a = b (since a and b can take values from o to p – 1); hence number of such  matrices is p
and when a + b = multiple of p ⇒ a, b can take p – 1 values. (the pairs
(1, p – 1), (2, p – 2), (3, p – 3) and so on
∴ Total number of matrices = p + p – 1
= 2p – 1.
If a matrix A is skew symmetric then A = – A^T, which, in this case, is possible only if a = o and b = o. That gives a null matrix which has already been counted once in skew symmetric case.
Hence, total number of matrix is still 2p – 1