What is the Invertibility of 2N-I Matrix When N^2 = N?

In summary, to show that the square matrix 2N - I is its own inverse if N^{2} = N, we can use the properties of invertible matrix and calculate (2N-I)(2N-I). By expanding this expression and using N^2=N, we can see that the desired result, (2N-I)(2N-I) = I, is achieved.
  • #1
Precursor
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Homework Statement


Show that the square matrix [tex]2N - I[/tex] is its own inverse if [tex]N^{2} = N[/tex]

Homework Equations


properties of invertible matrix


The Attempt at a Solution


I really don't know where to start here. I know that [tex](2N-I)(2N-I) = I[/tex], but where do I go on from there?
 
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  • #2
You are correct in calculating (2N-I)(2N-I) and using N2=N in your calculations. If (2N-I)(2N-I)=I, then the desired result is achieved, as this implies [(2N-I)]-1=(2N-I).
 
  • #3
VeeEight said:
You are correct in calculating (2N-I)(2N-I) and using N2=N in your calculations. If (2N-I)(2N-I)=I, then the desired result is achieved, as this implies [(2N-I)]-1=(2N-I).

So the answer is simply [tex](2N-I)(2N-I) = I[/tex]? But how did I use [tex]N^{2} = N[/tex]?
 
  • #4
I just assumed you did :)
The result (2N-I)(2N-I)=I is the desired result. So take (2N-I)(2N-I) and expand it. After you are done that, use N2=N.
 
  • #5
VeeEight said:
I just assumed you did :)
The result (2N-I)(2N-I)=I is the desired result. So take (2N-I)(2N-I) and expand it. After you are done that, use N2=N.

ok, so I get:

[tex]2N^{2} - 2NI - 2NI + I^{2} = I[/tex]

[tex]2N - 2N - 2N + I = I[/tex]

But this doesn't work out.
 
  • #6
The coefficient on N2 is incorrect. It is not a 2.
 
  • #7
VeeEight said:
The coefficient on N2 is incorrect. It is not a 2.

Ok I got it. Thanks for the help.
 
  • #8
Cheers.
 

1. What is an invertible square matrix?

An invertible square matrix is a square matrix (a matrix with the same number of rows and columns) that has an inverse matrix, meaning that when multiplied together, they result in the identity matrix (a square matrix with 1s on the main diagonal and 0s everywhere else).

2. How do you determine if a square matrix is invertible?

A square matrix is invertible if its determinant (the value that represents how the matrix scales the area of a shape) is not equal to 0. If the determinant is 0, then the matrix does not have an inverse.

3. What is the significance of an invertible square matrix?

An invertible square matrix is significant because it allows for the solution of systems of linear equations. It also has a number of other applications in fields such as physics, engineering, and computer science.

4. Can a non-square matrix be invertible?

No, a non-square matrix cannot be invertible. In order for a matrix to have an inverse, it must be a square matrix.

5. How do you find the inverse of a given invertible square matrix?

To find the inverse of an invertible square matrix, you can use the Gauss-Jordan elimination method or the adjugate matrix method. Both methods involve manipulating the original matrix through a series of steps to ultimately arrive at the inverse matrix.

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