Matrices and Determinants 1 Question 19
19. The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both and $\operatorname{det}(A)$ is divisible by $p$ is
(a) $(p-1)^{2}$
(b) $2(p-1)$
(c) $(p-1)^{2}+1$
(d) $2 p-1$
NOTE: The trace of a matrix is the sum of its diagonal entries.
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Answer:
Correct Answer: 19. (d)
Solution:
- Given, $A=\begin{bmatrix}a & b \\ c & a\end{bmatrix}$, $a, b, c \in{0,1,2, \ldots, p-1}$
If $A$ is skew-symmetric matrix, then $a=0, b=-c$
$\therefore \quad|A|=-b^{2}$
Thus, $P$ divides $|A|$, only when $b=0 \quad$…(i)
Again, if $A$ is symmetric matrix, then $b=c$ and
$ |A|=a^{2}-b^{2} $
Thus, $p$ divides $|A|$, if either $p$ divides $(a-b)$ or $p$ divides $(a+b)$.
$p$ divides $(a-b)$, only when $a=b$,
i.e. $\quad a=b \in{0,1,2, \ldots,(p-1)}$
i.e. $p$ choices $\quad$…(ii)
$p$ divides $(a+b)$
$\Rightarrow p$ choices, including $a=b=0$ included in Eq. (i).
$\therefore$ Total number of choices are $(p+p-1)=2 p-1$
