The matrix
\[\mathbf{A} = \begin{pmatrix} 4 & 7 \\ c & d \end{pmatrix}\]has the property that if you are told the value of $\|\mathbf{v}\|,$ then you can derive the value of $\|\mathbf{A} \mathbf{v}\|.$  Compute $|c - d|,$ assuming that $c$ and $d$ are real numbers.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The matrix
\[\mathbf{A} = \begin{pmatrix} 4 & 7 \\ c & d \end{pmatrix}\]has the property that if you are told the value of $\|\mathbf{v}\|,$ then you can derive the value of $\|\mathbf{A} \mathbf{v}\|.$  Compute $|c - d|,$ assuming that $c$ and $d$ are real numbers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
11
The matrix
\[\mathbf{A} = \begin{pmatrix} 4 & 7 \\ c & d \end{pmatrix}\]has the property that if you are told the value of $\|\mathbf{v}\|,$ then you can derive the value of $\|\mathbf{A} \mathbf{v}\|.$  Compute $|c - d|,$ assuming that $c$ and $d$ are real numbers.
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