If a matrix is obtained from A by elementary transformations then .

Proof: The only thing that matters whether determinants equal zero or not.

Interchanging two rows or two columns of the matrix changes the sign of a determinant.

Multiplying a row (column) by a nonzero number multiplies a determinant by that number.

Adding a row (column) to another one holds the magnitude of a determinant.

Therefore, all singular submatrices transform into singular submatrices, and nonsingular submatrices transform into nonsingular submatrices.

Hence, the theorem.