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Math Help - invertible matrix

  1. #1
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    invertible matrix

    Let A be an nxn matrix (not assumed to be invertible). Show that there exists an invertible nxn matrix P such that
    APA = A

    if such matrix existed then wouldn't it mean PA = I ( nxn identity matrix) and
    AP = I hence A is invertible.
    but anyways here are my ideas so far on this.
    I am guessing P would be an elementary matrix (i.e. the ones concerned with row / column operations) if you left multiple by these elementary matrices you are doing row operations and if you right multiply you are doing column operations.
    any sort of helpful hints would be much appreciated.
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  2. #2
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    Quote Originally Posted by bubble86 View Post
    Let A be an nxn matrix (not assumed to be invertible). Show that there exists an invertible nxn matrix P such that
    APA = A

    if such matrix existed then wouldn't it mean PA = I ( nxn identity matrix) and
    AP = I hence A is invertible.
    but anyways here are my ideas so far on this.
    I am guessing P would be an elementary matrix (i.e. the ones concerned with row / column operations) if you left multiple by these elementary matrices you are doing row operations and if you right multiply you are doing column operations.
    any sort of helpful hints would be much appreciated.
    This problem is much easier if you think in terms of linear transformations rather than matrices.

    Let \{f_1, f_2,\ldots,f_r\} be a basis for the range of A, where f_i = Ae_i\;\,(1\leqslant i\leqslant r). Then we can extend \{e_1,\ldots,e_r\} to a basis of \mathbb{R}^n by adding vectors e_{r+1},\ldots,e_n say, and we can extend \{f_1,\ldots,f_r\} to a basis of \mathbb{R}^n by adding vectors f_{r+1},\ldots,f_n. Now define P to be (the matrix of) the linear transformation defined by Pf_i = e_i\;\,(1\leqslant i\leqslant n). Then P is invertible, and APA = A.
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