Question

The number of A in $T_P$ such that the trace of A is not divisible by p but det (A) is divisible by p is [Note: The trace of a matrix is the sum of its diagonal entries.]

• A. $(p-1)(p^2-p+1)$
• B. $p^3-(p-1{)}^2$
• C. $(p-1{)}^2$
• D. $(p-1)(p^2-2)$

Answer 1

The correct option is C $(p-1{)}^2$

Trace of $A=2a$ will be divisible by p iff $a=0$

$\left|A\right|=a^2-bc$

for $(a^2-bc)$ to be divisible by $(p-1)$ ordered pairs $(b,c)$ for any value of A

$\therefore$Required no. is ${(p-1)}^2$

Option C