Rank of an $n\times n$ matrix after deleting a row and a column from it.

Question

Let $M$ be a matrix with $n$ columns and $n$ rows, $\text{rank}(M) =n$, $B$ be a matrix generated from $M$ by deleting one row and one column of $M$. Then $\text{rank}(B) \ge n-2$, am I right?

Answer 1

Since $\text{rank}(M)=n$, all $n$ rows are linearly independent. So after deleting a row, there are $n-1$ rows which are linearly independent, and hence the rank becomes $n-1$. By the definition of the rank, this is also the maximum of the number of linearly independent columns. Thus deleting a column leaves at least $n-2$ linearly independent columns in the resulting matrix. So indeed the final rank is $\geq n-2$.

As an example where the equality holds, consider the matrix $\begin{pmatrix}1&1\\1&0\end{pmatrix}$ and deleting the first row and the first column.

Also, as an example where the inequality is strict, consider the identity matrix of rank $n$. After deleting the first row and the first column, the rank becomes $n-1>n-2$.