Consider the following n times n square matrix Show that aj1

Consider the following n times n square matrix. Show that a_j1 C_i1 + a_j2 C_i2 +...+ a_jn C_in = 0, where a_jk denotes the k^th element in the j^th row, C_il denotes the cofactor of the l^th element in the j^th row2, and j notequalto i. A=[a_11 a_1n a_n1 a_nn]

Solution

Consider the alien expansion which uses the entries of j-th row
aj1, ..., ajn and cofactors of i-th row Ci1, ..., Cin
aj1 · Ci1 + aj2 · Ci2 + ... + ain · Cin.
This is Laplas expansion of the matrix
.
A\' = ( a11 ...............a1n
............................
aj1 ................ajn
       ...........................
       ..........................
       aj1....................ajn
       ............................
       aj1......................ajn
       .............................
       .............................
       an1.......................ann
        )
           with two equal rows, thus det(A\') = 0.