Is A Identical if A is a Square Matrix with Full Rank?

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In summary, the conversation discusses proving that if a $m \times m$ matrix $A$ has rank $m$ and satisfies the condition $A^2=A$, then it will be identical. Various attempts are made to reach this conclusion, such as considering the linear independence and using $A^{-1}$, but it is ultimately determined that stating the property that a matrix of rank $m$ is invertible may suffice.
  • #1
evinda
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Hello! (Wave)

I want to prove that if a $m \times m$ matrix $A$ has rank $m$ and satisfies the condition $A^2=A$ then it will be identical.

$A^2=A \Rightarrow A^2-A=0 \Rightarrow A(A-I)=0$.

From this we get that either $A=0$ or $A=I$.

Since $A$ has rank $m$, it follows that it has $m$ non-zero rows, and so it cannot be $0$.

Thus $A=I$. Is everything right? Or could something be improved? (Thinking)
 
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  • #2
evinda said:
Hello! (Wave)

I want to prove that if a $m \times m$ matrix $A$ has rank $m$ and satisfies the condition $A^2=A$ then it will be identical.

$A^2=A \Rightarrow A^2-A=0 \Rightarrow A(A-I)=0$.

From this we get that either $A=0$ or $A=I$.

Hey evinda!

I don't think we can draw that conclusion.
Consider $A=\begin{pmatrix}1&0\\1&0\end{pmatrix}$, what is $A(A-I)$? (Worried)
 
  • #3
Klaas van Aarsen said:
Hey evinda!

I don't think we can draw that conclusion.
Consider $A=\begin{pmatrix}1&0\\1&0\end{pmatrix}$, what is $A(A-I)$? (Worried)

It is $0$ although neither $A$ nor $A-I$ is $0$.

How else can we get to the conclusion that $A=I$ ? (Thinking)

Do we use the linear independence? (Thinking)
 
  • #4
evinda said:
It is $0$ although neither $A$ nor $A-I$ is $0$.

How else can we get to the conclusion that $A=I$ ?

Do we use the linear independence?

How about multiplying by $A^{-1}$?
$A$ is invertible isn't it? (Thinking)
 
  • #5
Klaas van Aarsen said:
How about multiplying by $A^{-1}$?
$A$ is invertible isn't it? (Thinking)

So is it as follows? (Thinking)

Since the rank of the $m\times m$ matrix $A$ is $m$ we have that at the echelon form the matrix has no zero-rows. This implies that at the diagonal all elements are different from zero and this implies that the determinant is different from zero, since the determinant is equal to the product of the diagonal elements. Since the determinant is not equal to zero, the matrix is invertible. So $A^{-1}$ exists.

Thus $A^2-A=0 \Rightarrow A(A-I)=0 \Rightarrow A^{-1}A(A-I)=0 \Rightarrow A-I=0 \Rightarrow A=I$.

Is it complete now? Could we improve something? (Thinking)
 
  • #6
evinda said:
So is it as follows?

Since the rank of the $m\times m$ matrix $A$ is $m$ we have that at the echelon form the matrix has no zero-rows. This implies that at the diagonal all elements are different from zero and this implies that the determinant is different from zero, since the determinant is equal to the product of the diagonal elements. Since the determinant is not equal to zero, the matrix is invertible. So $A^{-1}$ exists.

Thus $A^2-A=0 \Rightarrow A(A-I)=0 \Rightarrow A^{-1}A(A-I)=0 \Rightarrow A-I=0 \Rightarrow A=I$.

Is it complete now? Could we improve something?

All good. (Nod)

It may suffice to simply state that an $m\times m$ matrix of rank $m$ is invertible though.
It's a property of matrices:
If A is a square matrix (i.e., m = n), then A is invertible if and only if A has rank n (that is, A has full rank).
(Nerd)
 
  • #7
Klaas van Aarsen said:
All good. (Nod)

It may suffice to simply state that an $m\times m$ matrix of rank $m$ is invertible though.
It's a property of matrices:
If A is a square matrix (i.e., m = n), then A is invertible if and only if A has rank n (that is, A has full rank).
(Nerd)

I see... Thanks a lot! (Party)
 

What does it mean to "show that A is identical"?

The phrase "show that A is identical" typically refers to proving that two things are exactly the same. This could involve demonstrating that they have the same characteristics, properties, or qualities.

What are the steps to showing that A is identical?

The steps to showing that A is identical may vary depending on the specific context and what is being compared. However, some general steps may include identifying the characteristics or properties of A, comparing them to those of another object, and providing evidence or reasoning to support the claim that they are identical.

How do you use scientific methods to show that A is identical?

In order to show that A is identical using scientific methods, it is important to use a systematic and objective approach. This may involve conducting experiments, collecting data, and analyzing results in order to support the claim that A is identical to another object.

Can you use mathematical equations to show that A is identical?

Yes, mathematical equations can be a useful tool in showing that A is identical to something else. By using equations to compare and quantify the characteristics or properties of A and another object, you can provide clear and precise evidence to support the claim of identicalness.

Why is it important to show that A is identical?

Demonstrating that A is identical to something else can be important for a variety of reasons. It can help us better understand the nature of A, provide evidence for theories or hypotheses, and contribute to the overall body of scientific knowledge. Additionally, showing that A is identical can also have practical applications, such as in the development of new technologies or treatments.

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