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All the entries of the matrix \( \begin{pmatrix} 5 & 2 \\ c & -7 \end{pmatrix} \)and its inverse are integers. Find all possible values of \({c}\).

Here's what I have so far:

I know that the inverse of matrix \( \begin{pmatrix} 5 & 2 \\ c & -7 \end{pmatrix} \) is\({\frac{1}{-35-2c}}\) \( \begin{pmatrix} -7 & -2 \\ -c & 5 \end{pmatrix} \), given the inverse formula. However, I'm not quite sure where to go from this. I tried distributing the discriminant part, but I didn't really get anything from that. Any tips or hints on how to proceed? Thanks!

 Mar 12, 2021
edited by Guest  Mar 12, 2021
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I took your working and looked for integer answers.

 


\(\frac{1}{-35-2c} \begin{pmatrix} -7 & -2 \\ -c & 5 \end{pmatrix}\\\)

 

each of these elements must be an integer.  so

 

\(\frac{-2}{-35-2c}\)   must be an integer,    I only looked at -2 because the absolute value of it is the smallest one.

 

so    \( -2\le -35-2c \le 2\\ 33\le -2c \le 37\\ 16.5\le -c \le 18.5\\ -16.5\ge c \ge -18.5\\ -18.5\le c \le -16.5\\ \)

but c is an integer, so c might be   -18 or   -17

 

I checked both of these and they both worked.

 Mar 13, 2021

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