Proof that In^-1=In | Linear Algebra

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SUMMARY

The discussion confirms that the inverse of the identity matrix \( I_n \) is indeed \( I_n \) itself, as demonstrated through the definition of matrix inverses. Specifically, the proof utilizes the property that \( A^{-1}A = I_n \) and \( AA^{-1} = I_n \). In this case, substituting \( A \) with \( I_n \) shows that \( I_n^{-1} = I_n \), validating the assertion. This conclusion is supported by the fundamental definition of matrix inverses in linear algebra.

PREREQUISITES
  • Understanding of matrix operations, specifically multiplication.
  • Familiarity with the definition of the inverse of a matrix.
  • Knowledge of identity matrices and their properties.
  • Basic concepts of linear algebra.
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This discussion is beneficial for students of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of matrix inverses and identity matrices.

cleopatra
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Homework Statement



In^-1=In
proof that!

Homework Equations


1 0
0 1
= I2^-1= I2 for an example.
 
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The inverse matrix A^{-1} of A is by definition the matrix such that A^{-1}A=I_n and AA^{-1}=I_n. So is I_n the inverse of I_n?
 
yes In is the inverese of In because In^-1 is the inverse of In and In^-1=In
true?
 
anyone?
 
cleopatra said:
yes In is the inverese of In because In^-1 is the inverse of In and In^-1=In
true?

Just use the definition. You want to check that the inverse of I_n is I_n itself (this is just another way of saying I_n^{-1}=I_n). What it comes down to is that I_nI_n=I_n.
 

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