Linear Algebra Matrix Proof

In summary, the conversation discusses proving or disproving a statement about matrices, specifically regarding the invertibility of (A-cI) when c is not equal to 1 or 2. The conversation explores the use of eigenvalues and eigenvectors to solve the problem, ultimately using a proof by contradiction to show that A-cI is indeed invertible when c is not equal to 1 or 2.
  • #1
11
0

Homework Statement



Prove or disprove the following statements. I and 0 denote respectively the identity and zero matrix of the same size as A. If A is a square matrix such that A^2 - 3A +2I = 0 then A-cI is invertible whenever c is not equal to 1 and c is not equal to 2.

Homework Equations



The Attempt at a Solution



I have factored the function to: (A-2I)(A-I)=0.

However, we can't assume that A=2I and A=I because we are dealing with matrices (i.e. two non zero matrices can produce the zero matrix when multiplied together). I have a feeling that Eigenvectors might be related to this question, but I don't know how to apply that concept in this scenario.

Thanks for your help.
 
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  • #2
mreaume said:

Homework Statement



Prove or disprove the following statements. I and 0 denote respectively the identity and zero matrix of the same size as A. If A is a square matrix such that A^2 - 3A +2I = 0 then A-cI is invertible whenever c is not equal to 1 and c is not equal to 2.

Homework Equations



The Attempt at a Solution



I have factored the function to: (A-2I)(A-I)=0.

However, we can't assume that A=2I and A=I because we are dealing with matrices (i.e. two non zero matrices can produce the zero matrix when multiplied together). I have a feeling that Eigenvectors might be related to this question, but I don't know how to apply that concept in this scenario.

Thanks for your help.

Do you have the Cayley-Hamilton theorem available?
 
  • #3
Yes, think about eigenvalues indirectly. Suppose A-cI is not invertible. That means there is a nonzero vector v such that (A-cI)v=0. So Av=cv. Can you use that to show (A^2-3A+2I)v=0 can only be true if c=2 or c=1?
 
Last edited:
  • #4
LCKurtz, I do not have the Cayley-Hamilton theorem available. It has not been presented in class yet.

Dick, I have understood the following:

(A-cI)v=0
So Av=cv
I imagine that we need det(A-cI)=0 in order to prove that A-cI is invertible.

I am stuck here. I don't know how to relate A^2 -3A +2I to the previous equations.
 
  • #5
Go with Dick's suggestion. If Av = cv, try to evaluate (A2 - 3A + 2I)v, which you know has to be zero by the statement of the problem.
 
  • #6
Ok. Here is what I have:

(A^2 -3A +2I) = 0
A^2v -3Av +2Iv = 0
AAv -3Av + 2v = 0
(c^2)v - 3cv + 2v = 0
(c^2 - 3c + 2)v = 0
((c-2)(c-1))v=0

Therefore (A-cI) is not invertible when c=1 c=2.

I think I understand now. Does everything seem to be in order in my procedure?

Thanks for your help.
 
  • #7
mreaume said:
Ok. Here is what I have:

(A^2 -3A +2I) = 0
A^2v -3Av +2Iv = 0
AAv -3Av + 2v = 0
(c^2)v - 3cv + 2v = 0
(c^2 - 3c + 2)v = 0
((c-2)(c-1))v=0

Therefore (A-cI) is not invertible when c=1 c=2.

I think I understand now. Does everything seem to be in order in my procedure?

Thanks for your help.

Yes, that's it. The important point is that (c-2)(c-1)v=0 is ONLY possible if c=1 or c=2, since v is nonzero.
 
  • #8
Perfect. Thanks for your help!
 
  • #9
mreaume said:
Ok. Here is what I have:

(A^2 -3A +2I) = 0
A^2v -3Av +2Iv = 0
AAv -3Av + 2v = 0
(c^2)v - 3cv + 2v = 0
(c^2 - 3c + 2)v = 0
((c-2)(c-1))v=0

Therefore (A-cI) is not invertible when c=1 c=2.
Your last statement is wrong- you are asserting that "(A-cI) is not invertible when c=1 c=2" when what you want to prove is "A- cI is invertible as long as long as [itex]c\ne 1[/itex] and [itex]c\ne 2[/itex]", the converse. Did you mean to say "(A-cI) is not invertible only when c=1 c=2"?

I think I understand now. Does everything seem to be in order in my procedure?

Thanks for your help.
 
  • #10
HallsofIvy said:
Your last statement is wrong- you are asserting that "(A-cI) is not invertible when c=1 c=2" when what you want to prove is "A- cI is invertible as long as long as [itex]c\ne 1[/itex] and [itex]c\ne 2[/itex]", the converse. Did you mean to say "(A-cI) is not invertible only when c=1 c=2"?
OP, you should definitely not take this as excessively pedantic. What you have written, though somewhat correct in the vernacular language, is incorrect in terms of mathematical formalism.
 
  • #11
So then how *can* you show that (A-cI) **is** invertible when c /= 1, 2?
 
  • #12
raxAdaam said:
So then how *can* you show that (A-cI) **is** invertible when c /= 1, 2?
It is shown. What he has said is different from that.
 
  • #13
Sorry "/= 1,2" is meant as "not equal to 1, 2" which, as HallsofIvy pointed out, is not proven.
 
  • #14
raxAdaam said:
Sorry "/= 1,2" is meant as "not equal to 1, 2" which, as HallsofIvy pointed out, is not proven.

It is if you the organize the thoughts correctly. We assumed A-cI is NOT invertible and concluded that then there is a nonzero vector such that (c-1)(c-2)v=0. That's a contradiction unless c=1 or c=2. Hence if c is not equal to 1 or 2 then A-cI IS invertible. Treat it as a proof by contradiction.
 
  • #15
Thanks Dick - don't know why I missed that; guess I got disoriented in the semantics. Thanks for clarifying it explicitly :D.
 

1. What is a matrix in linear algebra?

A matrix in linear algebra is a rectangular array of numbers or symbols arranged in rows and columns. It is used to represent a system of linear equations and perform various mathematical operations such as addition, subtraction, and multiplication.

2. What is a proof in linear algebra?

A proof in linear algebra is a logical and systematic explanation of how a mathematical statement or theorem is true. It involves using the properties and principles of linear algebra to show that a given statement is valid.

3. How is linear algebra used to prove matrix equations?

Linear algebra is used to prove matrix equations by manipulating the elements of a matrix using properties such as associativity, distributivity, and commutativity. These operations are used to transform one side of the equation into the other, thus proving their equality.

4. What are some common techniques used in matrix proofs?

Some common techniques used in matrix proofs include direct proof, proof by contradiction, and proof by induction. These techniques involve using logical reasoning and mathematical properties to show that a statement is true.

5. How do matrix proofs relate to real-world applications?

Matrix proofs have various real-world applications, such as in computer graphics, data analysis, and optimization problems. They are used to solve systems of linear equations and make predictions based on patterns and relationships between variables.

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