Mathematical Thinking - Matrices

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SUMMARY

The discussion centers on constructing a counterexample to the statement "for matrices A with real entries, A^3 = Identity implies A = Identity." A key counterexample provided is the rotation matrix $\begin{pmatrix}\cos\varphi&-\sin\varphi\\\sin\varphi&\cos\varphi\end{pmatrix}$, which represents a rotation by angle $\varphi$ in the plane. This matrix satisfies the condition A^3 = Identity for specific angles, demonstrating that A does not necessarily equal the Identity matrix.

PREREQUISITES
  • Understanding of matrix operations and properties
  • Familiarity with rotation matrices in two-dimensional space
  • Knowledge of matrix identities and eigenvalues
  • Basic concepts of linear algebra
NEXT STEPS
  • Study the properties of rotation matrices in detail
  • Explore the implications of matrix powers and identities
  • Learn about eigenvalues and eigenvectors of matrices
  • Investigate other types of counterexamples in linear algebra
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Students of linear algebra, mathematicians exploring matrix theory, and educators seeking to illustrate concepts of matrix identities and counterexamples.

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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.
 

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