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## Max Dimension for a linear space of singular matrices

Denote by $V$ a vector space of singular ($\det =0$) $n \times n$ matrices. What is the maximal dimension of such a space.

Denote by $W$ a vector space of $n \times n$ matrices with rank smaller or equal to $p < n$. What is the maximal dimension of such a space?

Answers: For the first question $n^2-n$ and for the second question $np$. I will present proofs soon. The first problem is an obvious subcase of the second problem. For a nice proof, look at the answer given in the next link: http://math.stackexchange.com/questions/66877/max-dimension-of-a-subspace-of-singular-n-times-n-matrices