The number of A in Tp such that det(A) is not divisible by p is [2010]

Question

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]

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

Smart Achievers · Accepted Answer

(4)

The number of matrices for which $T_{p}$ 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 divide $| A |$ are ${(p - 1)}^{2} .$

$∴$ Required number $\det (A)$

Solution

Q.
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]

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

1 $2 p^{2}$

2 $p^{3} - 5 p$

3 $p^{3} - 3 p$

4 $p^{3} - p^{2}$

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Solution

Q. Let p be an odd prime number and Tp be the following set of 2×2 matrices: Tp={A=[abca]: a,b,c∈{0,1,2,…,p-1}} [2010] Q. The number of A in Tp such that det(A) is not divisible by p is

1 2p2

2 p3-5p

3 p3-3p

4 p3-p2