Introduction

In this article, we explore the number of matrices in the set T_p whose determinant is divisible by an odd prime number p. We start by defining the set and the conditions under which the determinant is divisible by p. We then proceed through a detailed analysis, breaking down the problem into manageable cases and utilizing properties of modular arithmetic to arrive at the final answer.

Define the Set and General Matrix

Let p be an odd prime number and consider the set T_p of matrices of the form:

[ A begin{bmatrix} a b c a end{bmatrix} ]

The determinant of matrix A is given by:

[ text{det}(A) a^2 - bc ]

We seek the number of matrices in T_p where the determinant det(A) is divisible by p, i.e., a^2 - bc equiv 0 pmod{p} or a^2 equiv bc pmod{p}.

Case Analysis

Case 1: (a 0)

In this case, the equation simplifies to:

[ 0 equiv bc pmod{p} ]

This condition is satisfied if either b 0 or c 0 or both. Let's count the number of matrices for each subcase:

For b 0, c can take any value from 0 to p-1, giving p choices. For c 0, b can take any value from 0 to p-1, giving another p choices.

However, the case where b 0 and c 0 is counted twice, so we subtract 1. The total number of matrices for (a 0) is:

[ p p - 1 2p - 1 ]

Case 2: (a eq 0)

For values of a from 1 to p-1, there are p-1 choices. We need to solve:

[ bc equiv a^2 pmod{p} ]

For each fixed a, (a^2) is a specific value in {0, 1, ..., p-1}. We need to count the number of pairs (b, c) such that bc equiv a^2 pmod{p}.

The number of pairs (b, c) that satisfy bc equiv k pmod{p} is:

p when k 0, as either b 0 or c 0 gives p solutions. p for each k eq 0 due to the fact that any non-zero k can be expressed as a product of two elements from {0, 1, ..., p-1}.

Thus for each non-zero a, there are p pairs (b, c) that satisfy the equation. Since there are p-1 choices for a, the total number of matrices for (a eq 0) is:

[ (p-1) times p ]

Total Count

Combining both cases, the total number of matrices in T_p whose determinant is divisible by p is:

[ (2p - 1) times (p-1) times p (2p - 1) times (p^2 - p) ]

Simplifying, we get:

[ p^2 times (p - 1) ]

Thus, the final answer is:

[ boxed{p^2 times (p - 1)} ]