Determinant of an $n×n$ identity matrix with two arbitrary rows

Question

It seems true that the determinant should be the two rows with numbers and take from it only two columns where there aren't 1s in their columns. But I can't prove it's true.

Answer 1

Computing directly that determinant, if the entries as $a_{ij}$ where for $j>2$, then calling the matrix $A$ and the determinant of the attached matrices: $$|A|= a_{11}\cdot A_{11} + a_{12}\cdot A_{12}$$ Being the first two rows the rows that are changed. We now have that $A_{11}=a_{22}$ and $A_{12}=a_{21}$. So finally the determinant of that matrix is the cross multiplication or that four elements: $$|A| = a_{11}\cdot a_{22}+ a_{12}\cdot a_{21}$$