How to prove that the sum of the products of the elements of any column (row) of n-order determinant d = / AIJ / and the corresponding elements of another column (row) is equal to zero

How to prove that the sum of the products of the elements of any column (row) of n-order determinant d = / AIJ / and the corresponding elements of another column (row) is equal to zero

Let d be the determinant, and let the two lines indicated in the title be the i-th and j-th lines respectively
|a11 ... a1n|
|... |
|ai1 ... ain|
D= |... |=aj1Aj1+...+ajnAjn
|aji ... ajn|
|... |
|an1 ... ann|
If it is replaced by another line of elements and multiplied, ai1aj1 +... Ainajn = | a11... A1N|
|... |
|ai1 ... ain|
|... |
|ai1 ... ain|
|... |
|This is from the meaning of the title
Obviously, the line aj1... AJN is replaced by Ai1... Ain, that is Aji = Ai1,... AJN = ain. In this way, two lines in the determinant are the same, so the determinant value is 0