The matrix a is 13 by 91. give the smallest possible dimension for nul
a.

Use the rank-nullity theorem. It says that the rank of a matrix , , has the following relationship with its nullity and its number of columns :



We're given that is , i.e. has columns. The largest rank that a matrix can have is ; in this case, that would be 13.

So if we take to be of rank 13, i.e. we maximize its rank, we must simultaneously be minimizing its nullity, so that the smallest possible value for is given by