Show that det(A) is the product of all the roots of the characteristic

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SUMMARY

The determinant of an n x n matrix A is definitively the product of all the roots of its characteristic polynomial. The characteristic polynomial is derived from the eigenvalues of matrix A, which can be represented in either diagonal or Jordan form. In both representations, the determinant is calculated as the product of the eigenvalues located on the main diagonal. This establishes a clear relationship between the determinant and the roots of the characteristic polynomial.

PREREQUISITES
  • Understanding of matrix theory, specifically n x n matrices
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with the concept of characteristic polynomials
  • Comprehension of matrix similarity transformations (diagonal and Jordan forms)
NEXT STEPS
  • Study the properties of eigenvalues in linear algebra
  • Learn about the derivation and applications of characteristic polynomials
  • Explore matrix similarity and its implications on determinants
  • Investigate the proof techniques for determinant properties in linear algebra
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and anyone interested in understanding the relationship between determinants and eigenvalues.

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Let A be an n x n matrix. Show that det(A) is the product of all the roots of the characteristic polynomial of A.
 
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State you definitions of det(A) and the characteristic polynomial of A, and show what you have done so far. If your definition of det(A) is the product of all the roots of the characteristic polynomial of A, you are done.
 


The quick proof is that any matrix is similar to either a diagonal matrix or a Jordan matrix with its eigenvalues on the main diagonal. In either case, the determinant is just the product of the numbers on the main diagonal- the product of the eigenvalues.

The difficulty with you not showing any attempt yourself is that we don't know what definitions and theorems you have available to prove this.
 

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