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!

Guest Mar 12, 2021

edited by
Guest
Mar 12, 2021

#2**+1 **

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.

Melody Mar 13, 2021