# Question: hey everyone got this as an assignment question and i...

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Hey everyone,

Got this as an assignment question and I honestly have no idea where to start. If someone could point me in the right direction, that would be great.

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-angled triangles whose sides have whole number lengths. The simplest and most well-known example is (3, 4, 5).

(a) Let (a, b, c) be a Pythagorean triple and let P = $\left[\begin{array}{cc}c+b& a\\ a& c-b\end{array}\right]$ . Verify that P has all the following properties:

A: All entries of P are positive integers.

B: P is symmetric (this means P^T = P).

C: det P = 0.

D: p11 − p22 = 2k for some integer k > p12 (pij is the i, j−th entry of P.)

(b) Conversely, verify that if a 2×2 matrix P has properties A–D then P = $\left[\begin{array}{cc}c+b& a\\ a& c-b\end{array}\right]$ for a suitable Pythagorean triple (a, b, c).

Thanks in advance!