Let D= | array ccc n & n^2 & n^3 n^2 & n^3 & n^5 1 & 2 & 3 array | , then _ n M₁₁+C₃₃ (M₁₃ )^2 is equal to (where M_ i j is the minor and C_ i j is the cofactor of i^ th row and j^ th column's element)

Step-by-step solution

D= | array ccc n & n^2 & n^3 n^2 & n^3 & n^5 1 & 2 & 3 array | Then, M₁₁= | array cc n^3 & n^5 2 & 3 array |=3 n^3-2 n^5 and C₃₃= | array cc n & n^2 n^2 & n^3 array |=n^4-n^4=0 aligned M₁₃ & = | array cc n^2 & n^3 1 & 2 array |=2 n^2-n^3 _ n & = M₁₁+C₃₃ (M₁₃ )^2 = _ n n^3 (3-2 n^2 )+0 n^4(2-n)^2 & = _ n 3-2 n^2 n [4+n^2-4 n ] = _ n n^2 ( 3 n^2 -2 ) n^3 ( 4 n^2 +1- 4 n ) aligned put n= = ( 3 ^2 -2 ) ( 4 ^2 +1- 4 ) = -2 1 =0