Linear Algebra Proof: Invertible Idempotent Matrix Must be Identity Matrix

Just because AB = I doesn't mean that A = I and B = I. For example, let A = [1 0, 0 0] and B = [1 0; 0 0]. Then AB = I but A and B are not equal to I.
  • #1

Homework Statement


If A is an invertible idempotent matrix, then A must be the Identity matrix I_n.

Homework Equations


A^2==A ; A^2==AA; A^(-1); I==A^(-1)

The Attempt at a Solution



Suppose A is an nxn matrix =/= I_n.

s.t. A^(2)==A

so A^(2)==A ==> AA==A

==> A^(-1)AA==A^(-1)A ==> A==I==> A^(-1)A==A^(-1)I==>I==A^(-1)I==A^(-1)==A

which yeilds a contradiction because we supposed our A =/= I_n.

Therefore A==I_nIs this correct please help me understand where I have failed...
 
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  • #2
Dosmascerveza said:
Suppose A is an nxn matrix =/= I_n.

s.t. A^(2)==A

so A^(2)==A ==> AA==A
==> A^(-1)AA==A^(-1)A
==> A==I
You should have stopped right here. You should have also stated that A is invertible.
==> A^(-1)A==A^(-1)I
==> I==A^(-1)I==A^(-1)==A
This is wrong. You don't know that the inverse of A is equal to A.
 
  • #3
Okay so if i stated A an invertible nxn matrix =/= I_n

s.t A^(2)==A(idempotent)... truncating the last bit of foolishness. I was correct?
 
  • #4
another proof...
problem statement.
prove if A and B are idempotent and AB==BA then AB is idempotent.

AB==BA ==> A^(-1), B^(-1) exist

Since A and B are idempotent invertible matrices, from previously proven theorem, we know A=I and B=I. and since II==I ==> AB==I Therefore AB==BA and AB is Idempotent,
 
  • #5
Dosmascerveza said:
AB==BA ==> A^(-1), B^(-1) exist
This isn't true.
 

1. What is a linear algebra proof?

A linear algebra proof is a process of using mathematical techniques and logical reasoning to demonstrate the validity of a statement or theorem related to linear algebra. It involves manipulating matrices, vectors, and other mathematical objects to show that a given statement is true.

2. What is an invertible idempotent matrix?

An invertible idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. Additionally, it has an inverse matrix, meaning that it can be multiplied by another matrix to give the identity matrix. In other words, an invertible idempotent matrix is its own inverse.

3. Why must an invertible idempotent matrix be the identity matrix?

This is because an invertible idempotent matrix, by definition, has an inverse matrix. The only matrix that is its own inverse is the identity matrix, which is a square matrix with 1's along the main diagonal and 0's everywhere else. Therefore, an invertible idempotent matrix must be the identity matrix.

4. How can a linear algebra proof demonstrate that an invertible idempotent matrix must be the identity matrix?

A linear algebra proof can demonstrate this by using the properties of invertible and idempotent matrices. By manipulating the equations and properties, the proof will show that the only matrix that satisfies both of these properties is the identity matrix. This proves that any invertible idempotent matrix must be the identity matrix.

5. What are some real-world applications of the concept of invertible idempotent matrices?

Invertible idempotent matrices have various applications in fields such as computer science, physics, and engineering. In computer graphics, they are used to rotate and scale images. In physics, they are used to describe the quantum states of particles. In engineering, they are used to model electrical circuits and control systems. Additionally, they are used in cryptography to encrypt and decrypt data.

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