Matrices and Determinants 1 Question 18
18. The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $\operatorname{det}(A)$ is divisible by $p$ is
(a) $(p-1)\left(p^{2}-p+1\right)$
(b) $p^{3}-(p-1)^{2}$
(c) $(p-1)^{2}$
(d) $(p-1)\left(p^{2}-2\right)$
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Answer:
Correct Answer: 18. (c)
Solution:
- Trace of $A=2 a$, will be divisible by $p$, iff $a=0$.
$|A|=a^{2}-b c$, for $\left(a^{2}-b c\right)$ to be divisible by $p$.
There are exactly $(p-1)$ ordered pairs $(b, c)$ for any value of $a$.
$\therefore$ Required number is $(p-1)^{2}$.
