Question

The number of A in T${}_p$ such that the trace of A is not divisible by p but det(A) divisible by p is ?[Note: The trace of 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 $p$ iif $a=0$
$\left|A\right|=a^2-bc$
For $(a^2-bc)$ to be divisible by $p$.
There are exactly $(p-1)$ ordered pairs $(b,c)$ for any value of $a$
Therefore required number is ${(p-1)}^2$