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

  • Thread starter Thread starter evinda
  • Start date Start date
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
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)
 
Physics news on Phys.org
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)
 
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)
 
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)
 
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)
 
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)
 
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)
 

Similar threads

Back
Top