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Given the matrix:

$$ \begin{pmatrix} 0 & 0 & 1 & 1\\ -2 & 2 & 3 & -3\\ 1 & -1 & -2 & 1 \\ \end{pmatrix} $$

  1. Find an invertible matrix P and a Reduced row echelon form matrix D such that: $ PA = D $
  2. Find a bases for $ColA$
  3. Find a second matrix $Q$ that: $P\neq Q$ and $QA = D$

I've managed to solve questions 1: $$ D = \begin{pmatrix} 1 & -1 & 0 & 3\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 \\ \end{pmatrix} , P = \begin{pmatrix} 0 & -2 & -3\\ 0 & -1 & -2\\ 1 & 1 & 2 \\ \end{pmatrix} $$

and question 2: $$ \begin{bmatrix} 0 \\ 2 \\ 1 \\ \end{bmatrix}, \begin{bmatrix} 1 \\ 3 \\ -2 \\ \end{bmatrix}$$

But I don't know how to solve question 3.

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Hint -

$Q = A^{-1}D$

You already found D and find inverse of A. Then multiply them. You will get Q.

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