Sum of invertible matrices is invertible?

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SUMMARY

The sum of two invertible matrices is not necessarily invertible. A counterexample is provided using 2x2 matrices, where two invertible matrices, such as the scalars 2 and -2, yield a non-invertible result when summed. This highlights that while individual matrices may be invertible, their sum can lead to a determinant of zero, indicating non-invertibility.

PREREQUISITES
  • Understanding of matrix operations, specifically addition and multiplication.
  • Knowledge of invertible matrices and their properties.
  • Familiarity with determinants and their significance in matrix invertibility.
  • Basic linear algebra concepts, including vector spaces.
NEXT STEPS
  • Explore the properties of determinants in relation to matrix addition.
  • Study examples of non-invertible matrices and their sums.
  • Learn about the conditions under which the sum of matrices remains invertible.
  • Investigate the implications of matrix rank on invertibility.
USEFUL FOR

Students of linear algebra, mathematics educators, and anyone studying matrix theory or seeking to understand the properties of invertible matrices.

soul5
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Homework Statement



Is the sum of two invertible matrices necessarily invertible?


The Attempt at a Solution



I don't know please help.
 
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Try a couple of small examples - use [tex]2 \times 2[/tex]
matrices that you know are invertible.

One small hint: both [tex]2[/tex] and [tex]-2[/tex] are invertible real numbers, but their sum isn't invertible.
 

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