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Describe the set of 3×3 matrices where the sum of the entries in any given row or column adds up to 1. (By expressing all entries of such matrix in terms of a few parameters).

For example, the matrix $\left[\begin{array}{l}1&2&-2\\3&5&-7\\-3&-6&10\end{array}\right]$

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    What do you mean by "describe the set"? Isn't calling it *"the set of 3×3 matrices where the sum of the entries in any given row or column adds up to 1"* describing the set? – 2017-01-16
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    I meant to express all entries of such matrix in terms of a few parameters – 2017-01-16
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    Then that's what you should have said. Do you have any thoughts on the problem? Have you tried anything on your own here? – 2017-01-16
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    This is all i got so far$\left[\begin{array}{l}a&b&1-a-b\\c&d&1-c-d\\1-c-a&1-d-b&a+b+c+d-1\end{array}\right]$ – 2017-01-16
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    so, what's wrong with that? Is your question about whether you can use fewer than $4$ parameters? – 2017-01-16
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    I was just wondering if that is enough to describe the set. The question stated a few parameters, when i tried I wasn't able to use fewer than 4. – 2017-01-16
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    I would certainly say that it's enough. Clearly, as soon as you know those four parameters, all other entries are forced. On the other hand, for **any** combination of values for those parameters, there's a valid matrix. That is, our parametrization over $\Bbb R^4$ is both injective and surjective. – 2017-01-16
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    A tricky aspect of this might be **proving** that it's impossible to do this with fewer than $4$ parameters if we use a linear or continuous parameterization. There exist theorems that do that for you. – 2017-01-16
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    That helped a lot. Thank you so much. – 2017-01-16
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    Glad to do it. When asking questions in the future, try to make it clearer what exactly the issue is. – 2017-01-16

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We can express any such matrix as $$ \pmatrix{ a_{11} & a_{12} & 1 - a_{11} - a_{12}\\ a_{21} & a_{22} & 1 - a_{21} - a_{22}\\ 1 - a_{11} - a_{21} & 1 - a_{12} - a_{22} & a_{11} + a_{12} + a_{21} + a_{22} - 1 } $$

From the discussion in the comments: why is this parameteriation a "good description" of the set? As soon as you know those four parameters, all other entries are forced. On the other hand, for any combination of values for those parameters, there's a valid matrix. That is, our parametrization over $\Bbb R^4$ is both injective and surjective.