Matrices & Determinants
Matrices and Determinants
Grade Class 12
Question:
The number of A in T<sub>p</sub> such that the trace of A is not divisible by p but det (A) is divisible by p is -<br>[Note: The trace of a matrix is the sum of its diagonal entries.]
(p - 1) (p<sup>2</sup> - p + 1)
p<sup>3</sup> - (p - 1)<sup>2</sup>
(p - 1)<sup>2</sup>
(p - 1) (p<sup>2</sup> - 2)
Step-by-Step Solution
Key Concept: The problem involves counting matrices over a finite field Z_p (or T_p) satisfying specific conditions on trace and determinant. This requires combinatorial counting of matrices with given trace and determinant properties.
The total number of 2x2 matrices over Z_p is p^4. The condition involves trace and determinant. The number of matrices with trace t and determinant d is known to be p^2 - 1 if t^2 - 4d is a non-zero square, p^2 + p if t^2 - 4d = 0, and p^2 - p if t^2 - 4d is a non-square. Summing over the conditions given leads to the result (p-1)^2.
Correct Answer: 3