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 February 12th, 2013, 12:27 PM #1 Newbie   Joined: Feb 2013 Posts: 2 Thanks: 0 Find matrix M such that AM=B Find matrix M such that AM=B when A = 1 -2 3 2 1 7 2 2 7 B = 0 0 1 1 0 1 0 1 1 0 1 0 I think side by side Gaussian Elimination is used somehow.
 February 12th, 2013, 12:50 PM #2 Member   Joined: Jan 2013 Posts: 93 Thanks: 0 Re: Find matrix M such that AM=B Can you compute $A^{-1}$ and multiply $B$ on the left by that?
 February 12th, 2013, 01:04 PM #3 Newbie   Joined: Feb 2013 Posts: 2 Thanks: 0 Re: Find matrix M such that AM=B Yeah thanks
February 12th, 2013, 02:07 PM   #4
Math Team

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Re: Find matrix M such that AM=B

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Quote:
 $\text{Find matrix }M\text{ such that }AM\,=\,B\,\text{ when:}$ [color=beige]. . [/color]$A\:=\:\begin{bmatrix}1&-2=&3 \\ \\ 2=&1=&7 \\ \\ 2=&2=&7 \end{bmatrix}\;\;\;B \:=\:\begin{bmatrix}0&1=&1 \\ \\ 0=&1=&1 \\ \\ 1=&1=&0 \end{bmatrix}=$

$\text{Find }A^{-1}.$

$\;\;\;\text{W\!e have: }\:\left[\begin{array}{ccc|ccc} 1&-2&3&1&0&0 \\ \\ 2&1&7 & 0&1&0 \\ \\ 2&2&7 & 0&0&1 \end{array}\right]$

$\;\;\;\;\begin{array}{c}\\ \\ \\ \\ \\ R_2-2R_1 \\ R_3-2R_1 \end{array}\:\left[\begin{array}{ccc|ccc}1&-2&3&1&0&0 \\ \\ \\ 0&5&1&-2&1&0 \\ \\ \\ 0&6&1&-2&0&1 \end{array}\right]$

$\begin{array}{c}\\ \\ -(R_2-R_3) \\ \\ \end{array}\;\left[\begin{array}{ccc|ccc}1&-2&3&1&0&0 \\ \\ 0&1&0 & 0&-1&1 \\ \\ 0&6&1&-2&0&1 \end{array}\right]$

$\;\;\begin{array}{c}R_1+2R_2 \\ \\ \\ \\ \\ \\ R_3-6R_2 \end{array}\;\left[\begin{array}{ccc|ccc}1&0&3&1&-2&2 \\ \\ 0&1&0&9&-1&1 \\ \\ 0&0&1&-2&6&-5 \end{array}\right]$

$\;\;\;\begin{array}{c}R_1-3R_3 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}\;\left[\begin{array}{ccc|ccc}1&0&0&7&-20&17 \\ \\ 0&1&0&0&-1&1 \\ \\ 0&0&1&-2&6&-5 \end{array}\right]$

$\text{Hence: }\:A^{-1} \;=\;\begin{bmatrix}7&-20=&17 \\ 0=&-1=&1 \\ -2=&6=&-5\end{bmatrix}=$

$\text{Hence: }\:M \;=\;A^{-1}\,\cdot\,B \;=\;\begin{bmatrix}7&-20=&17 \\ 0=&-1=&1 \\ -2=&6=&-5\end{bmatrix}\:\begin{bmatrix}0=&1=&1 \\ 0=&1=&1 \\ 1=&1=&0 \end{bmatrix}=$

$\text{Therefore: }\:M \;=\;\begin{bmatrix}17&-20=&-24=&-13 \\ \\ \\ 1=&-1=&1=&-1 \\ \\ \\ -5=&6=&-7=&4 \end{bmatrix}=$

 February 15th, 2013, 12:16 PM #5 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Find matrix M such that AM=B Same idea but a little faster: Replace the identity matrix on the left by the matrix B: $\begin{bmatrix}1 & -2 & 3 \\ 2 & 1 & 7 \\ 2 & 2 & 7 \end{bmatrix}\begin{bmatrix}0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 \end{bmatrix}$ and do the row operations, as Soroban shows, to reduce the matrix, A, to the identity while applying those same row operations to the matrix, B.

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