I tought a matrix that can be row reduced to the identity matrix is also invertible.

Question

This matrix i can convert the identity matrix. $\begin{bmatrix} 1 &2 &-1 \\ -1 &0 &3 \\ 0 &1 &1 \end{bmatrix}.$

see here:

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still, the determinant is 0 of the first matrix. I thought if a matrix has 3 pivot points then it is invertible?

Was my tought wrong or is there something wrong with this matrix? I thought it was a theorem that if a matrix has n pivot points it is automatically invertible

Answer 1

The error lies in the assumption that the determinant is $0$. Actually, it is equal to $-2$.