Search any question & find its solution
Question:
Answered & Verified by Expert
Paragraph:
Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices
$$
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
$$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.]
Options:
Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices
$$
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
$$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.]
Solution:
1926 Upvotes
Verified Answer
The correct answer is:
$(p-1)^2$
$(p-1)^2$
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 $(p-1)^2$.
Hence, (c) is the correct option.
$|A|=a^2-b c$, 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 $(p-1)^2$.
Hence, (c) is the correct option.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.