# Inverting all values in matrix

Lets say I have a matrix: $$\left[\begin{array}{cc} 2 & 4 \\ 3 & 7 \\ \end{array}\right]$$

And my maximum range value is $10$, how would I go about creating another matrix that inverts those values? So that the matrix would end up looking like:
$$\left[\begin{array}{cc} 8 & 6 \\ 7 & 3 \\ \end{array}\right]$$ In algebraic form?

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What is the maximum range value? – 1015 Mar 8 '13 at 2:07
What do you mean by "maximum range value"? Which "values" are you talking about inverting? – Cameron Buie Mar 8 '13 at 2:07
I've edited your question to use $\LaTeX$. Please make sure it still represents your original intent. For help with formatting in the future, please see this meta question. – apnorton Mar 8 '13 at 2:28

Given $m>0$, is there some way to transform $$\left[\begin{array}{cc}a & b\\c & d\end{array}\right]\mapsto\left[\begin{array}{cc}m-a & m-b\\m-c & m-d\end{array}\right]$$ for all $a,b,c,d$ between $0$ and $m$ (inclusive)?

The answer to that question is: "Yes, but it isn't necessarily very interesting." Let $J$ be the $2\times 2$ matrix of $1$s. Then for any $m>0$ and any $2\times 2$ matrix $A$ with entries between $0$ and $m$ (inclusive), the matrix $m\cdot J-A$ does the trick (where $m\cdot J$ indicates scalar multiplication by $m$).

If that's not what you were trying to ask, then please clarify. It might help if you told us what led you to ask this question, too.

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It may not be interesting mathematically, but it can do some pretty interesting stuff to a black-and-white photograph... :) – apnorton Mar 8 '13 at 2:34
@anorton: Fair point...at least, once generalized to bigger matrices. I've yet to see an interesting $4$-pixel photo. ;-) – Cameron Buie Mar 8 '13 at 2:36
But I'm sure that a 4-pixel photo could garner millions at a modern art museum! :P Seriously, that is a fair point as well... I wasn't thinking about the size of the matrix at all. – apnorton Mar 8 '13 at 2:40

Look at the individual entries in the matrix, and see what relationship you can find between those entries, and the "maximum range value".

From your example, in the $(1,1)$ cell in the first matrix, you have the entry $2$. In the second matrix, you have the entry $8$. Your maximum value is $10$. Can you see some relationship between the numbers $2$, $8$, and $10$? Does this relationship hold for the other values? (for example, $3$, $7$, and $10$)

Leave a comment if you need more guidance. (I'm experimenting with my answering style from just giving an answer to trying a more Socratic method... so I'm looking for feedback.)

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so they would both sum to 10... does that mean if i had a matrix full of 10's and then subtracted the original matrix from that i would get the inverse :D.. I think thats what i was looking for! – Taylor Dale Mar 8 '13 at 2:42
@TaylorDale yep, that's the idea! Just a side-note: if you're writing up anything about this, you may want to be careful with the term "inverse"--it has a different meaning for matrices than you may think. :) – apnorton Mar 8 '13 at 2:46