Matrix Inverse Problem

**MEMPHIS** · October 10th 2010, 01:13 PM

Problem:
Solve the matrix equation for $X$
$\begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix} X_(2,3) \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 1 \\ 4 & -1 & 1 \end{pmatrix}$

Attempt at a Solution:
I started working on this problem thinking I could take the inverse of both the matrices being multiplied by $X_2,3$ and then multiply each one by the matrix on the right hand side. Well that worked for the 2,2 matrix, but then I remembered you can't multiply a 3,3 matrix and a 2,3 matrix. I know I could transpose the 2,3 matrix to a 3,2 matrix so I could multiply it by the 3,3 matrix, but how would this affect my answer? Also is this the way to approach this problem or is there an easier way? All advice/help is appreciated!

**pickslides** · October 10th 2010, 01:31 PM

Assuming you mean X is to be a 2x3 matrix then follow this

$AXB=C$

$A^{-1}AXB=A^{-1}C$

$XBB^{-1}=A^{-1}CB^{-1}$

$X=A^{-1}CB^{-1}$

Therefore

$X_{(2,3)} = \begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix}^{-1} \begin{pmatrix} 1 & 2 & 1 \\ 4 & -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}^{-1}$

**MEMPHIS** · October 10th 2010, 01:39 PM

Oh okay, so that way I don't have to transpose the 2,3 matrix. Awesome! Thanks very much!

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