In my book this property has been given without proof:
Sum of the products of the elements of any row or column with the cofactors of the corresponding elements of any other row or column is $0$.
Can someone please provide the proof of it? I would prefer a proof without the i,j and summation notation. They are just hard to interpret and counter-intuitive.
Let's say the row of expansion is the $i$-th and the row of multiplication is the $j$-th. By definition, you are calculating the determinant of a matrix $A'$ that is equal to $A$, except for the $i$-th row, which is equal to the $j$-th. Since $A'$ has two equal lines, $\det A'=0$.
Same for columns.