Is I-A Non-Singular? Proving Non-Singularity and Inverse for a Matrix

In summary: That makes sense. Thanks for clearing that up for me.In summary, the question asks for a proof that the matrix I-A is non-singular. TheAttempt at a Solution provides an outline of how to do this, but it is not self-evident.
  • #1
ahsanxr
350
6

Homework Statement



Let A be an n x n matrix. Show that if A2=0 then I - A is non-singular and (I - A)-1= I + A

Homework Equations





The Attempt at a Solution



Ok, so I started off with finding the general form of a 2x2 matrix which when squared gives a zero matrix, and all the properties above are satisfied. But how do I show that for an n x n matrix? Please help me out.
 
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  • #2
What is (I-A)*(I+A)?
 
  • #3
The Identity matrix.
 
  • #4
ahsanxr said:
The Identity matrix.

Hence? What do you conclude from that?
 
  • #5
But I only know that because the question says to show it and the "experiment" I did with my 2x2 matrix shows that. I cannot prove it. Please give a more thorough explanation :)
 
  • #6
ahsanxr said:
But I only know that because the question says to show it and the "experiment" I did with my 2x2 matrix shows that. I cannot prove it. Please give a more thorough explanation :)

If A*B=I then A=B^(-1). Isn't that the definition of inverse?
 
  • #7
Yes, obviously. I do not know how that is connected to the question though. It asks us to show that the inverse of I-A is I+A which at this point is not self-evident (or at least I think so).
 
  • #8
ahsanxr said:
Yes, obviously. I do not know how that is connected to the question though. It asks us to show that the inverse of I-A is I+A which at this point is not self-evident (or at least I think so).

Now you are just plain confusing me. If (I-A)*(I+A)=I then as I read the definition of 'inverse' that means (I-A)^(-1)=(I+A). If you mean showing its the inverse by using the explicit calculation you probably used for 2x2 matrices, then you can't do that. You don't know what A is. It's still true though.
 
  • #9
Dick said:
(I-A)*(I+A)=I

How? This is my question.
 
  • #10
ahsanxr said:
How? This is my question.

Multiply it out. (I+A)*(I-A)=? Use the distributive property! I thought you did that in post 3. I may have been wrong.
 
  • #11
Can you do that with matrices? I have read the property that (A+B)C=AC+BC but only to that extent.
 
  • #12
ahsanxr said:
Can you do that with matrices? I have read the property that (A+B)C=AC+BC but only to that extent.

Sure you can. (I+A)(I-A)=I(I-A)+A(I-A). Now distribute again.
 
  • #13
Yes, suppose C = D+E, hence AC+BC=(A+B)(D+E).
 
  • #14
Of course. That was kind of a stupid question. Thanks for clearing it up.

The question first asks us to show that I-A is non-singular. How do we show that? Or does that follow from (I-A)(I+A)=I and since I has an inverse, it must be non-singular?
 
  • #15
When two matrices are multiplied together to produce the identity, it means the two matrices are inverses of each other. What is the definition of a non-singular matrix?
 
  • #16
ahsanxr said:
Of course. That was kind of a stupid question. Thanks for clearing it up.

The question first asks us to show that I-A is non-singular. How do we show that? Or does that follow from (I-A)(I+A)=I and since I has an inverse, it must be non-singular?

You KNOW I has an inverse. It's I! It's doesn't follow from that. Please quote me the definition of what an inverse is?
 
  • #17
A matrix multiplied by the original matrix to give Identity.
 
  • #18
ahsanxr said:
A matrix multiplied by the original matrix to give Identity.

Ok, so (I+A)(I-A)=I. What's the inverse of (I-A)?
 
  • #19
I get that. That is what I'm asking. Is the proof of (I-A)'s non-singularity the fact that I+A is its inverse?
 
  • #20
ahsanxr said:
I get that. That is what I'm asking. Is the proof of (I-A)'s non-singularity the fact that I+A is its inverse?

Yes, yes, yes. If a matrix has an inverse, it's nonsingular. Now I'm going to have to ask to you look up the definition of 'nonsingular'.
 
  • #21
I know, I know. I was just verifying :P

But the question first asked us to prove the non-singularity and then the second part. Doesn't our approach do it the other way around?
 
  • #22
ahsanxr said:
I know, I know. I was just verifying :P

But the question first asked us to prove the non-singularity and then the second part. Doesn't our approach do it the other way around?

I don't think being asked to show a) and b) means you have to do them in that order. Or that you can't use the same expression to prove both.
 

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is used to represent and manipulate linear transformations and systems of linear equations.

2. What is a linear algebra proof?

A linear algebra proof is a mathematical argument or demonstration that uses the principles and properties of linear algebra to show the validity of a statement or theorem.

3. What are the basic properties of matrices?

The basic properties of matrices include addition, subtraction, multiplication, and scalar multiplication. Matrices also have properties such as associativity, commutativity, and distributivity.

4. How do you prove two matrices are equal?

To prove two matrices are equal, you must show that their corresponding elements are equal. This can be done by comparing each element in the same position in both matrices and showing that they are equal.

5. What is the purpose of using matrices in linear algebra?

Matrices are used in linear algebra to represent and solve systems of linear equations, perform operations on vectors, and represent linear transformations. They also have applications in computer graphics, economics, and other fields.

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