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Inverse of a matrix

  1. Jun 25, 2010 #1
    Let A be an invertible matrix.
    Then Ax=e1 will give us the first column of the inverse of A.
    Where e1 is the first column of the identity matrix.

    How can we prove this fact??:confused:
     
  2. jcsd
  3. Jun 25, 2010 #2

    HallsofIvy

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    From [itex]Ax= e_1[/itex] and the fact that A is invertible, we have [itex]A^{-1}Ax= x= A^{-1}e_1[/itex]. Now, can you convince yourself that any matrix times [itex]e_1[/itex] is the first column of the matrix? Try multiplying a few matrices times [itex]e_1[/itex] and see what happens:
    What is
    [tex]\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}[/tex]

    What is
    [tex]\begin{bmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}[/tex]
     
  4. Jun 25, 2010 #3
    OMG that was too simple thank u HallsofIvy :)
     
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