The nullity of a square matrix with linearly dependent rows is at least one. True or False?
Here is the answer my textbook gives:
True; if the rows are linearly dependent, then the rank is at least $1$ less than the number of rows, so since the matrix is square, its nullity is at least $1$.
I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
Best Answer
If the matrix is not square, then the matrix is non -invertible, and so the nullity is at least one.
More details: If your matrix is $m$ x $n$, and say $m>n$. We know that the rank of a matrix is equal to the row rank, which is equal to the column rank. So, rank of the matrix is at most $n$- the number of columns, which is less than $m$, and hence the nullity is at least $1$. Similar arguments apply for $m<n$.