1. ## Identity Matrix

I was wondering if anyone can agree or disagree with me.
If we consider square matrix.
If we use row reduced echelon form on square matrix.
I think that we can always bring the matrix into the form of identity matrix. for example. We can bring any square matrix (lets take 3x3 matrix as example) into this form.....

1 0 0
0 1 0
0 0 1

Same goes for any other square matrix.

2. False, consider following matrix;

$
\begin{bmatrix}
1 & -1 & 0 \\
1 & 1 & -1 \\
2 & 0 & -1
\end{bmatrix}
$

Reduced Row Echelon Form gives;

$
\begin{bmatrix}
1 & 0 & -.5 \\
0 & 1 & -.5 \\
0 & 0 & 0
\end{bmatrix}
$

3. Originally Posted by joker40
I was wondering if anyone can agree or disagree with me.
If we consider square matrix.
If we use row reduced echelon form on square matrix.
I think that we can always bring the matrix into the form of identity matrix. for example. We can bring any square matrix (lets take 3x3 matrix as example) into this form.....

1 0 0
0 1 0
0 0 1

Same goes for any other square matrix.
Not all matrices are nonsingular and singular matrices aren't row equivalent to I.

4. Originally Posted by joker40
I was wondering if anyone can agree or disagree with me.
If we consider square matrix.
If we use row reduced echelon form on square matrix.
I think that we can always bring the matrix into the form of identity matrix. for example. We can bring any square matrix (lets take 3x3 matrix as example) into this form.....

1 0 0
0 1 0
0 0 1

Same goes for any other square matrix.
A matrix $A_{n \times n}$ is only invertible if $|A| \neq 0$.