Mathematics CO-FACTOR AND ADJOINT

Minors :

Minors of an element is defined as the minor determinant obtained by deleting a particular row or colunm in which that element lies. e.g. in the determinant.

`D=|(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` minor of `a_12` denoted as

`M_12=|(a_21,a_23),(a_31,a_33)|`

`M_12=a_21 .a_33- a_31.a_23`

`M_11=|(a_22,a_23),(a_32,a_33)|`

`M_12=a_22 .a_33- a_32.a_23`

`M_13=|(a_21,a_22),(a_31,a_32)|`

`M_12=a_21 .a_32- a_31.a_22`

and so on for `M_21, M_22.........M_33`

`text(Important Results :)`

1. The sum of products of the elements of any row or column with their corresponding co factors is equal to the value of the determinant.

`Delta = a_(11)C_(11) + a_(12)C_(12) + a_(13)C_(13)`

`Delta = a_(11)C_(11) + a_(21)C_(21) + a_(31)C_(31)`

`Delta = a_(21)C_(21) + a_(22)C_(22) + a_(23)C_(23)`

`Delta = a_(12)C_(12) + a_(22)C_(22) + a_(32)C_(32)`

`Delta = a_(31)C_(31) + a_(32)C_(32) + a_(33)C_(33)`

`Delta = a_(13)C_(13) + a_(23)C_(23) + a_(33)C_(33)`

`Delta = |(a_11,a_12,a_13, ..., ...., a_1n),(a_21,a_22,a_23,....,.......,a_2n),(a_31,a_32,a_33,...,.....,a_3n),(...,..,..,..,...,....),(..,....,..,....,....,...),(a_(n1),a_(n2),a_(n3),...,....,a_(n n)) | = a_(11)C_(11) + a_(12)C_(12) + a_(13)C_(13) + ...........+ a_(1n)C_(1n)`

2. The sum of the product of element of any row (or column) with corresponding co-factors of another row (or column) is equal to zero.

`a_(11)C_(21) + a_(12)C_(22) + a_(13)C_(23) = 0`

`a_(11)C_(13) + a_(21)C_(23) + a_(31)C_(33) = 0`

3. If the value of an order determinant is .1., then the value of the determinant formed by the cofactors of corresponding elements of the given determinant is given by

`Delta^c = Delta^(n - 1 )`

i.e., in case of second order determinant

`Delta^c = Delta`

and third order determinant `Delta^c = Delta^2`

Cofactor :

It has no separate identity and is related to the cofactor as

`C_(ij)=(-1)^(i+j)M_(ij)` where `i` denotes the row and `j` denotes the column

Hence the value of a determinant `D=|(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` of order three in terms of `text(Minor)` and `text(Cofactor)` can be written as

`D = a_11 M_11 - a_12 M_12 + a_13 M_13` or

`= a_11 C_11 + a_12 C_12 + a_13 C_13`

`text(Note :)` Determinant of order `3` will have `9` minors and each minor will be a determinant of order `2` and a determinant of order `4` will have `16` minors and each minor will be determinant of order `3`.

Adjoint of a Square Matrix :

Let `A=[a_(ij)]=[(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)]` be a square matrix and let the matrix formed by the cofactors of `[a_(ij)]` in determiinant `|A|` is `=[(C_11,C_12,C_13),(C_21,C_22,C_23),(C_31,C_32,C_33)]`.

Then `(adj A) =[(C_11,C_12,C_13),(C_21,C_22,C_23),(C_31,C_32,C_33)]`

Hence the transpose of the matrix of cofactors of elements of `A` in `detA` is called the `adj A`.

`text( Properties of Adjoint :)`

`text(Property 1)` If A be a square matrix order n, then
`A(adj A)= (adj A)A =| A | I_n`
i.e., the product of a matrix and its adjoint is commutative.

`text(Deductions of Property 1)`
`text(Deduction 1)` If A be a square singular matrix of order n, then
`A(adj A)= (adj A) A= O` [null matrix]
Since, for singular matrix, `| A | =O.`
`text(Deduction 2)` If A be a square non-singular matrix of order n, then
`| adj A|= | A|^(n-1)`
Since, for non-singular matrix, `| A | ne O.`

`text(Property 2)` If A and Bare square matrices of order n, then
`adj(AB) = (adjB)(adjA)`

`text(Property 3)` If A is a square matrix of order n, then
`( adjA)' = adjA'`

`text(Property 4)` If A be a square non-singular matrix of order n, then
`adj (adjA) =| A|^(n- 2) A`

`text(Property 5)` If A be a square non-singular matrix of order n, then
`| adj (adj A) | = | A|^((n-1)^2)`

`text(Property 6)` If A be a square matrix of order n and k is a scalar, then
`adj (kA) = k^(n-1). (adj A)`