Mathematical Thinking - Matrices

AI Thread Summary
To construct a counterexample for the statement "for matrices A with real entries, A^3 = Identity implies A = Identity," consider the rotation matrix given by $\begin{pmatrix}\cos\varphi&-\sin\varphi\\\sin\varphi&\cos\varphi\end{pmatrix}$, which represents a rotation by angle $\varphi$. This matrix satisfies the condition A^3 = Identity for specific angles, such as $\varphi = \frac{2\pi}{3}$, yet it is not equal to the identity matrix. The discussion highlights the importance of exploring matrices beyond the identity to understand their properties. Counterexamples like rotation matrices demonstrate that A^3 can equal the identity without A being the identity itself. Understanding these nuances is crucial in mathematical thinking regarding matrices.
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Hi guys, iv been stuck on this problem for a while now and can't seem to make any headway

construct a counterexample to the following statement:

"for matrices A with real entries 'A^3=Identity impies A=Identity'

im not restricted by size for the matrix.

any hints would help because i just can't think of any counterexamples!

thankyou
 
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You may consider $\begin{pmatrix}\cos\varphi&-\sin\varphi\\\sin\varphi&\cos\varphi\end{pmatrix}$, which is the matrix of rotation on the plane by angle $\varphi$ around the origin.
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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