`color{red}{"If elements of a row (or column) are multiplied with cofactors of any
other row (or column), then their sum is zero."}`
For example,
Let `color {red} {Delta = [ (a_11 ,a_12 ,a _13 ),( a_21 , a_22 , a_23 ), ( a_31 ,a _32 , a_33 )]}`
` Δ = a_11 A_21 + a_12 A_22 + a_13 A_23`
`= a_11 (-1)^(1+1) | ( a_12, a_13), ( a_32, a_33) | + a_12 (-1)^(1+2) | (a_11 , a_13), ( a_31 , a_33) | + a_13 (-1)^(1+3) | (a_11, a_12), ( a_31, a_32) |`
`= | (a_11, a_12, a_13 ), ( a_11 , a_12, a_13) ,( a_31 ,a_32 , a_33) | = 0` (since `R_1` and `R_2` are identical)
The Sum of Product of element of any row (column) with Cofactor of other row (Column) is zero.
`color{blue} { a_11A_21 +a_12 A_22 + a_13 A_23 = 0}`
`color {blue} {a_21A_31 +a_22A_32 + a_23 A_33 = 0}`
`color {blue} {a_31A_11 + a_32 A_12 + a_33A_13 =0}`
Similarly, we can try for other rows and columns
`color{red}{"If elements of a row (or column) are multiplied with cofactors of any
other row (or column), then their sum is zero."}`
For example,
Let `color {red} {Delta = [ (a_11 ,a_12 ,a _13 ),( a_21 , a_22 , a_23 ), ( a_31 ,a _32 , a_33 )]}`
` Δ = a_11 A_21 + a_12 A_22 + a_13 A_23`
`= a_11 (-1)^(1+1) | ( a_12, a_13), ( a_32, a_33) | + a_12 (-1)^(1+2) | (a_11 , a_13), ( a_31 , a_33) | + a_13 (-1)^(1+3) | (a_11, a_12), ( a_31, a_32) |`
`= | (a_11, a_12, a_13 ), ( a_11 , a_12, a_13) ,( a_31 ,a_32 , a_33) | = 0` (since `R_1` and `R_2` are identical)
The Sum of Product of element of any row (column) with Cofactor of other row (Column) is zero.
`color{blue} { a_11A_21 +a_12 A_22 + a_13 A_23 = 0}`
`color {blue} {a_21A_31 +a_22A_32 + a_23 A_33 = 0}`
`color {blue} {a_31A_11 + a_32 A_12 + a_33A_13 =0}`
Similarly, we can try for other rows and columns