Matrix equation - solve for X

Mar 2016
1
0
Sweden
Hi everybody! I need some help with this matrix equation:

$$\pmatrix{6 & 1 \cr -3 & -8} -X \pmatrix{1 & 6 \cr -1 & 5}=I$$

So this is how I would solve it:

$$A-X\cdot B=I
\Leftrightarrow -X=I\cdot B^{-1}-A$$

Is this correct? Also, is -X different from X when it comes to matrices? Should I just see it as -1 times the matrix X, so then I would multiply all the numbers in the matrix with -1? I appreciate any help I can get with this!
 

Plato

MHF Helper
Aug 2006
22,507
8,664
matrix equation:
$$\pmatrix{6 & 1 \cr -3 & -8} -X \pmatrix{1 & 6 \cr -1 & 5}=I$$
\(\displaystyle X = \left( {\begin{array}{*{20}{c}}5&1\\{ - 3}&{ - 9}\end{array}} \right){\left( {\begin{array}{*{20}{c}}1&6\\{ - 1}&5\end{array}} \right)^{ - 1}}\)
 

Debsta

MHF Helper
Oct 2009
1,361
633
Brisbane
\(\displaystyle a - x . B = i \)
\(\displaystyle a - i = x . B \)
\(\displaystyle (a - i) . B^ -1 = x \)

Sorry, not too good at LaTex. They should all be capitals.
 
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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Hi everybody! I need some help with this matrix equation:

$$\pmatrix{6 & 1 \cr -3 & -8} -X \pmatrix{1 & 6 \cr -1 & 5}=I$$

So this is how I would solve it:

$$A-X\cdot B=I
\Leftrightarrow -X=I\cdot B^{-1}-A$$

Is this correct?
No, it's not. In fact, it would not be correct even if these were numbers. From a- xb= 1, multiplying by \(\displaystyle b^{-1}\) would give \(\displaystyle ab^{-1}- x=b^{-1}\) so that \(\displaystyle x= ab^{-1}- b^{-1}\). With A, B, X matrices, it is exactly the same with \(\displaystyle X= AB^{-1}- B^{-1}= (A- I)B^{-1}\).

Also, is -X different from X when it comes to matrices? Should I just see it as -1 times the matrix X, so then I would multiply all the numbers in the matrix with -1? I appreciate any help I can get with this!
-X is different from X in any algebraic system as long as X is not 0! Yes, you can interpret -X as -1 times X which would be all entries of X multiplied by -1. It is also the same as "X subtracted from the 0 matrix" as well as the diagonal matrix with all diagonal entries "-1" times X.
 
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