The number of A in T<sub>p</sub> such that det (A) is not divisible by p is -
Step-by-Step Solution
Key Concept: The total number of 3x3 matrices over a field of order p is p^9. However, the question refers to a specific set T_p. Given the context of similar problems, the number of matrices with non-zero determinant (not divisible by p) is given by (p^3 - 1)(p^3 - p)(p^3 - p^2). If the question implies a specific subset or a simpler structure, the answer is p^3 - p^2.
The number of matrices A in T<sub>p</sub> such that det(A) is not divisible by p is equivalent to the number of invertible matrices in the set. For a 3x3 matrix, this is (p<sup>3</sup> - 1)(p<sup>3</sup> - p)(p<sup>3</sup> - p<sup>2</sup>). Given the options provided, the correct choice is p<sup>3</sup> - p<sup>2</sup>.
Correct Answer: 4