Matrices And Determinants Ques 20

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) $(p-1)\left(p^{2}-2\right)$

Show Answer

Answer:

Correct Answer: 20.(c)

Solution:

  1. 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 non-zero value of $a$.

$\therefore$ Required number is $(p-1)^{2}$.

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