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 October 29th, 2010, 01:44 AM #1 Member   Joined: Apr 2010 Posts: 91 Thanks: 0 Matrix right inverse Find all right-inverses of the matrix $\begin{pmatrix}1 &-2 &3 \\ -4 & 5 &-6 \end{pmatrix}$ Attempt: $\mathbf{AR}= \mathbf{I_2}$ (A | R): $\begin{pmatrix}1 &-2 &3 \\ -4 & 5 &-6 \end{pmatrix}\begin{pmatrix}x_1 &y_1 \\ x_2&y_2 \\ x_3&y_3 \end{pmatrix}$ = $\begin{pmatrix}1 &0 \\ 0&1 \end{pmatrix}$ Now I tried to get the matrix into the form of (I | R) by gauss reduction: $\begin{pmatrix}1 &-2 &3& |1 &0 \\ -4 &5 &-6&|0&1 \end{pmatrix}$ but this will never work since the matrix on the left is 2 x 3 and not 2 x 2 and thus I will never get $\mathbf{I_2}$ on the left What can I do?
 October 29th, 2010, 03:17 PM #2 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Matrix right inverse Shootin in the dark, here... Write out what the actual elements of the product would be on the left hand side... $1*x_1 -2x_2 + 3x_3= 1$ since the matrix on the right hand side has a "1" in it's upper left hand. Now I'll write the lower left hand element... $-4x_1 + 5x_2 -6x_3= 0$ Add twice the first equation I wrote to the second... $-2x_1 + x_2= 2$ --> $x_2= 2x_1 + 2$ The process is similar for the y's. You solution will be any matrix of the form that you wrote, where $x_2= 2x_1 + 2$ and $x_1$is free. (Well, this, plus the similar result for the y's...)
 October 29th, 2010, 04:39 PM #3 Member   Joined: Apr 2010 Posts: 91 Thanks: 0 Re: Matrix right inverse The answer in my notes was: $\frac{1}{3}\begin{pmatrix}-5 &-2 \\ -4&-1 \\ 0&0 \end{pmatrix} + \begin{pmatrix}1\\ 2\\ 1\end{pmatrix}\begin{pmatrix}\lambda & \mu \end{pmatrix}$ where $\lambda,\mu\epsilon \mathbb{R}$, I am confused as I dont see the similarity to: $x_2= 2x_1 + 2$ and $y_2= 2y_1 + 2$
 October 29th, 2010, 09:51 PM #4 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Matrix right inverse If you sub those values for x2 and y2 back into the equations, you'll get something more easily comparable to the given solution

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### hiw to find right inverse of a matrix

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