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# Question: hello i think i am very close to finishing this...

###### Question details

Hello,

I think I am very close to finishing this question but am not quite there...

Question: A Pythagorean triple (a, b, c) consists of positive integers a, b and c satisfying a < b < c and ${a}^{2}+{b}^{2}={c}^{2}$. They correspond to right angles triangles whose sides have whole-number lengths. The simplest and most well known example is (3, 4, 5).

Show that if A is any 2 x 2 matrix and P is any matrix with properties B and C, then $AP{A}^{T}$ also has properties B and C.

The properties A-D (referred to above) are...

A: all entries of P are positive integers

B: P is symmetric (This means ${P}^{T}=P$).

C: det P =0

D: ${p}_{11}-{p}_{22}=2k$ for some integer k>${p}_{12}$ (${p}_{ij}$ is the i, j-th entry of P)

Click the link below to see what I've done already. (I can't upload images on here for some reason)

https://ibb.co/NtpFgpt

Thanks in advance!

###### Solution by an expert tutor 