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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.

Archer
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  • See also https://math.stackexchange.com/questions/1404189/why-in-the-proof-of-a-cdot-adja-deta-cdot-i-n-entires-not-on-the-diagonal . – darij grinberg Sep 18 '18 at 15:42

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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.