JEE Advanced · Mathematics · 18. Matrices
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.]
- A
\((p-1)\left(p^2-p+1\right)\)
- B
\(p^3-(p-1)^2\)
- C
\((p-1)^2\)
- D
\((p-1)\left(p^2-2\right)\)
Answer & Solution
Correct Answer
(C)
\((p-1)^2\)
Step-by-step Solution
Detailed explanation
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.
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