Write Minors and Cofactors of the elements of following determinants: (i) (ii)
(i)
The given determinant is,
| 2 − 4 0 3 |
Minor of an element a i j of a determinant obtained by deleting its i th row and j th column in which element a i j lies.
Minor of a 11 ,
M 11 = | 2 − 4 0 3 | = 3
Minor of a 12 ,
M 12 = | 2 − 4 0 3 | = 0
Minor of a 21 ,
M 21 = | 2 − 4 0 3 | = − 4
Minor of a 22 ,
M 22 = | 2 − 4 0 3 | = 2
Cofactor of an element is given by,
A i j = ( − 1 ) i + j M i j
Cofactor of a 11 ,
A 11 = ( − 1 ) 1 + 1 M 11 A 11 = ( − 1 ) 1 + 1 3 A 11 = 3
Cofactor of a 12 ,
A 12 = ( − 1 ) 1 + 2 M 12 A 12 = ( − 1 ) 3 × 0 A 12 = 0
Cofactor of a 21 ,
A 21 = ( − 1 ) 2 + 1 M 21 A 21 = ( − 1 ) 3 × ( − 4 ) A 21 = 4
Cofactor of a 22 ,
A 22 = ( − 1 ) 2 + 2 M 22 A 22 = ( − 1 ) 4 × 2 A 22 = 2
(ii)
The given determinant is,
| a c b d |
Minor of an element a i j of a determinant obtained by deleting its i th row and j th column in which element a i j lies.
Minor of a 11 ,
M 11 = | a c b d | = d
Minor of a 12 ,
M 12 = | a c b d | = b
Minor of a 21 ,
M 21 = | a c b d | = c
Minor of a 22 ,
M 22 = | a c b d | = a
Cofactor of an element is given by,
A i j = ( − 1 ) i + j M i j
Cofactor of a 11 ,
A 11 = ( − 1 ) 1 + 1 M 11 A 11 = ( − 1 ) 1 + 1 d A 11 = d
Cofactor of a 12 ,
A 12 = ( − 1 ) 1 + 2 M 12 A 12 = ( − 1 ) 3 × b A 12 = − b
Cofactor of a 21 ,
A 21 = ( − 1 ) 2 + 1 M 21 A 21 = ( − 1 ) 3 × c A 21 = − c
Cofactor of a 22 ,
A 22 = ( − 1 ) 2 + 2 M 22 A 22 = ( − 1 ) 4 × a A 22 = a
(i)
The given determinant is,
Minor of an element
Minor of
Minor of
Minor of
Minor of
Cofactor of an element is given by,
Cofactor of
Cofactor of
Cofactor of
Cofactor of
(ii)
The given determinant is,
Minor of an element
Minor of
Minor of
Minor of
Minor of
Cofactor of an element is given by,
Cofactor of
Cofactor of
Cofactor of
Cofactor of
