SATHEE: Matrices and Determinants 1 Question 17

Matrices and Determinants 1 Question 17

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17. The number of $A$ in $T_{p}$ such that $\det (A)$ is not divisible by $p$ , is

======= Passage II

Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices

$T_{p} = A = [\begin{matrix} a & b \\ c & a \end{matrix}]; a, b, c \in 0, 1, 2, \dots, p - 1$

(2010)

####17. The number of $A$ in $T_{p}$ such that $\det (A)$ is not divisible by $p$ , is

3e0f7ab6f6a50373c3f2dbda6ca2533482a77bed

(a) $2 p^{2}$

(b) $p^{3} - 5 p$

(d) $p^{3} - p^{2}$

Show Answer

Answer:

Correct Answer: 17. ( d)

Solution:

The number of matrices for which $p$ does not divide $Tr (A) = (p - 1) p^{2}$ of these $(p - 1)^{2}$ are such that $p$ divides $| A |$ . The number of matrices for which $p$ divides $Tr (A)$ and $p$ does not divides $| A |$ are $(p - 1)^{2}$ .

$∴$ Required number $= (p - 1) p^{2} - (p - 1)^{2} + (p - 1)^{2}$

$= p^{3} - p^{2}$

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Matrices and Determinants 1 Question 17

17. The number of A in Tp such that det(A) is not divisible by p, is

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17. The number of $A$ in $T_{p}$ such that $\det (A)$ is not divisible by $p$ , is