# Math Help - Determinantal rank

1. ## Determinantal rank

Let A be an n × n matrix. Define the determinantal rank detrankA to be the greatest value of r such that there is an r × r submatrix B of A (that is, a matrix B obtained by deleting n − r rows and n − r columns from A) such that detB $\neq$ 0. Show that detrankA = rankA.

Genuinally have no idea where to start, I can't even see that this is true so haven't got any ideas. Help appreciated!

2. The rank of a matrix is equal to the maximal number of linearly independent rows (and the same number of columns). If you delete the rest of the rows and columns, keeping this set of linearly independent ones, what can we say about the determinant of the matrix you have left over? (Consider what the reduced row-echelon form of the new matrix must look like, for instance. Or consider what the determinant says about invertibility of the matrix.)

Conversely, if you have a submatrix with nonzero determinant, can we say whether or not its rows (columns) must be linearly independent or not?