Write Minors and Cofactors of the elements of following determinants:
(i) `|(1, 0, 0),( 0, 1, 0),( 0, 0, 1)|` (ii) `|(1, 0, 4),( 3, 5,-1),( 0, 1, 2)|`

Explanation

(i) `|(1, 0, 0),( 0, 1, 0),( 0, 0, 1)|`

Minor of `a_(11)= M_(11)=|[1,0],[0,1]|=1(1)-0=1`

`a_(12)= M_(12)=|[0,0],[0,1]|=0-0=0`

`a_(13)= M_(13)=|[0,1],[0,0]|=0-0=0`

`a_(21)= M_(21)=|[0,0],[0,1]|=0-0=0`

`a_(22)= M_(22)=|[1,0],[0,1]|=1(1)-0=1`

`a_(23)= M_(23)=|[1,0],[0,0]|=0-0=0`

`a_(31)= M_(31)=|[0,0],[1,0]|=0-0=0`

`a_(32)= M_(32)=|[1,0],[0,0]|=0-0=0`

`a_(33)= M_(33)=|[1,0],[0,1]|=1(1)-0=1`


Cofactor of `a_(ij)=A_(ij)=(-1)^(i+j).M_(ij)`

Cofactor of `a_(11)=A_(11)=(-1)^(1+1).M_(11)=(-1)^2xx1=1`

`a_(12)=A_(12)=(-1)^(1+2).M_(12)=(-1)^3xx0=0`

`a_(13)=A_(13)=(-1)^(1+3).M_(13)=(-1)^4xx0=0`

`a_(21)=A_(21)=(-1)^(2+1).M_(21)=(-1)^3xx0=0`

`a_(22)=A_(22)=(-1)^(2+2).M_(22)=(-1)^4xx1=1`

`a_(23)=A_(23)=(-1)^(23).M_(23)=(-1)^5xx0=0`

`a_(31)=A_(31)=(-1)^(3+1).M_(31)=(-1)^4xx0=0`

`a_(32)=A_(32)=(-1)^(3+2).M_(32)=(-1)^5xx0=0`

`a_(33)=A_(33)=(-1)^(3+3).M_(33)=(-1)^6xx1=1`



(ii) `|(1, 0, 4),( 3, 5,-1),( 0, 1, 2)|`

Minor of `a_(11)= M_(11)=|[5,-1],[1,2]|=5(2)-1(-1)=10+1=11`

`a_(12)= M_(12)=|[3,-1],[0,2]|=6-0=6`

`a_(13)= M_(13)=|[3,5],[0,1]|=3-0=3`

`a_(21)= M_(21)=|[0,4],[1,2]|=0-4=-4`

`a_(22)= M_(22)=|[1,4],[0,2]|=2-0=2`

`a_(23)= M_(23)=|[1,0],[0,1]|=1-0=1`

`a_(31)= M_(31)=|[0,4],[5,-1]|=0-20=-20`

`a_(32)= M_(32)=|[1,4],[3,-1]|=-1-12=-13`

`a_(33)= M_(33)=|[1,0],[3,5]|=5-0=5`


Cofactor of `a_(ij)=A_(ij)=(-1)^(i+j).M_(ij)`

Cofactor of `a_(11)=A_(11)=(-1)^(1+1).M_(11)=(-1)^2xx11=11`

`a_(12)=A_(12)=(-1)^(1+2).M_(12)=(-1)^3xx6=(-1)6=-6`

`a_(13)=A_(13)=(-1)^(1+3).M_(13)=(-1)^4xx3=(1)3=3`

`a_(21)=A_(21)=(-1)^(2+1).M_(21)=(-1)^3xx(-4)=(-1)(-4)=4`

`a_(22)=A_(22)=(-1)^(2+2).M_(22)=(-1)^4xx2=(1)(2)=2`

`a_(23)=A_(23)=(-1)^(23).M_(23)=(-1)^5xx1=(-1)(1)=-1`

`a_(31)=A_(31)=(-1)^(3+1).M_(31)=(-1)^4xx(-20)=1(-20)=-20`

`a_(32)=A_(32)=(-1)^(3+2).M_(32)=(-1)^5xx(-13)=(-1)(-13)=1
`

`a_(33)=A_(33)=(-1)^(3+3).M_(33)=(-1)^6xx(5)=1(5)=5`