Proof that In^-1=In | Linear Algebra

  • Thread starter cleopatra
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In summary, the conversation discusses the proof that In is its own inverse. Using the definition of an inverse matrix, it is shown that In^-1=In, making the statement true. The conversation also emphasizes that this can be checked by the fact that I_nI_n=I_n.
  • #1
cleopatra
45
0

Homework Statement



In^-1=In
proof that!

Homework Equations


1 0
0 1
= I2^-1= I2 for an example.
 
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  • #2
The inverse matrix [tex]A^{-1}[/tex] of [tex]A[/tex] is by definition the matrix such that [tex]A^{-1}A=I_n[/tex] and [tex]AA^{-1}=I_n[/tex]. So is [tex]I_n[/tex] the inverse of [tex]I_n[/tex]?
 
  • #3
yes In is the inverese of In because In^-1 is the inverse of In and In^-1=In
true?
 
  • #4
anyone?
 
  • #5
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 [tex]I_n[/tex] is [tex]I_n[/tex] itself (this is just another way of saying [tex]I_n^{-1}=I_n[/tex]). What it comes down to is that [tex]I_nI_n=I_n[/tex].
 

1. What is In^-1?

In^-1 is the inverse of the identity matrix, In. It is a square matrix that, when multiplied by In, results in the identity matrix In. In other words, it "undoes" the effects of In.

2. How is the inverse of a matrix calculated?

The inverse of a matrix is calculated by using a mathematical process called the Gaussian elimination method. This involves manipulating the rows and columns of the matrix to reduce it to a specific form, from which the inverse can be easily determined.

3. Why is the inverse of the identity matrix equal to itself?

The identity matrix, In, is a special type of matrix where all the diagonal elements are equal to 1 and all other elements are equal to 0. When multiplied by its inverse, In^-1, the result is the identity matrix In. This is because In^-1 essentially "undoes" the effects of In, leaving us with the same matrix.

4. Can every matrix have an inverse?

No, not every matrix has an inverse. For a matrix to have an inverse, it must be a square matrix (i.e. have the same number of rows and columns) and must also meet certain mathematical conditions. If these conditions are not met, the matrix will not have an inverse.

5. How is the inverse of a matrix used in linear algebra?

The inverse of a matrix is an important tool in linear algebra. It is used to solve systems of linear equations, find solutions to matrix equations, and perform other operations such as finding determinants and calculating eigenvalues. It is also used in many real-world applications such as computer graphics, optimization problems, and engineering calculations.

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