Matrices and Determinants 1 Question 17
Passage II
Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices
$ T_{p}=A= \begin{bmatrix} a & b \\ c & a \end{bmatrix} ; a, b, c \in{0,1,2, \ldots, p-1} $
(2010)
17. The number of $A$ in $T_{p}$ such that $\operatorname{det}(A)$ is not divisible by $p$, is
(a) $2 p^{2}$
(c) $p^{3}-3 p$
(b) $p^{3}-5 p$
(d) $p^{3}-p^{2}$
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Answer:
Correct Answer: 17. ( d)
Solution:
- The number of matrices for which $p$ does not divide $\operatorname{Tr}(A)=(p-1) p^{2}$ of these $(p-1)^{2}$ are such that $p$ divides $|A|$. The number of matrices for which $p$ divides $\operatorname{Tr}(A)$ and $p$ does not divides $|A|$ are $(p-1)^{2}$.
$\therefore \quad$ Required number $=(p-1) p^{2}-(p-1)^{2}+(p-1)^{2}$
$ =p^{3}-p^{2} $
