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# Inverse of a Matrix

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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

### 1+0 Answers

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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