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Consider any nxn matrices A, B and C such that A^2 + B^2 = C^2 Then the matrix triple (A,B,C) is called a Matrix valued Pythagorean Triple.

I have observed that any nxn matrix M and N such that MN=NM, then I can extend Euclid's Fomula to generate a matrix Pythagorean triples, that is

A = M^2 - N^2 B = 2MN C = M^2 + N^2

Another method to generate Matrix Pythagorean triple was discussed by John D. Cook at his cite, johndcook.com. (im sorry, i dont know how to insert links)

However, the matrices A,B and C mention above are commutative to each other.

My question is how can I generate matrix valued Pythagorean triple (A,B,C) such that A,B and C do not commute to each other?

Another condition is that the matrices A,B and C must not be symmetric matrices

Here's one example; Let A = $$ \begin{bmatrix} 30 & 13 \\ 3 & 0 \\ \end{bmatrix} $$

B = $$ \begin{bmatrix} 4 & 8 \\ 12 & 16 \\ \end{bmatrix} $$

C = $$ \begin{bmatrix} -26 & -25 \\ -15 & 4 \\ \end{bmatrix} $$

Then A^2 + B^2 = C^2.

Please help me, ma'am and sir, to find a method to generate such noncommutative and nonsymmetric Matrix valued Pythagorean triples.

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    $\begingroup$ Solutions of $A^k + B^k = C^k$ in $ n x n$ integral matrices”, American Mathematical Monthly , 75, 1968, 759-760 $\endgroup$ – Ethan Bolker Sep 1 '18 at 16:34
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On Matrix Pythagorean Triples. American Mathematical Monthly, Volume 126, 2019, issue 2, pp 158- 160. Abstract We present a parametrization for the Pythagorean matrices, i.e., integer-valued matrices $A$, $B$, and $C$ such that $A^2+B^2=C^2$.

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