Question

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

Answer 1

Use the rank-nullity theorem. It says that the rank of a matrix $\mathbf A$, $\mathrm{rank}(\mathbf A)$, has the following relationship with its nullity $\mathrm{null}(\mathbf A)$ and its number of columns $n$:

$\mathrm{rank}(\mathbf A)+\mathrm{null}(\mathbf A)=n$

We're given that $\mathbf A$ is $13\times91$, i.e. has $n=91$ columns. The largest rank that a $m\times n$ matrix can have is $\min\{m,n\}$; in this case, that would be 13.

So if we take $\mathbf A$ 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 $\mathrm{null}(\mathbf A)$ is given by

$13+\mathrm{null}(\mathbf A)=91\implies\mathrm{null}(\mathbf A)=91-13=78$