`color{green} ✍️` The area of triangle whose vertices are `(x_1,y_1)` , `(x_2,y_2)` and `(x_3,y_3)`

`= |1/2[x_1(y_2 -y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)]|`

● Now this expression can be written in the form of a determinant as

`color{red}{"Area =" 1/2 |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1) |}`


Note : 1 . Area is a positive quantity, we always take the absolute value of the determinant.
2.`color{orange}{"The area of the triangle formed by three collinear points is zero"}`
3. If area is given, use both positive and negative values of the determinant for calculation.

`color{green} ✍️` Minor of an element `a_(ij)` of a determinant is the determinant obtained by deleting its `i^(th)` row and `j^(th)` column in which element `a_(ij)` lies. Minor of an element `a_(ij)` is denoted by `M_(ij)` .

`color {brown} text{Minor of Matrix }`

`color{red} {A = [ (a_11, a_12 , a_13 ), (a_21, a_22 , a_23 ),( a_31 ,a_32 ,a_33 )] }`

`M_11` = Minor of `color {blue} {a_11 = | (a_22, a_23), ( a_32 , a_33 ) |} = a_22 a_33 - a_23 a_32`

`M_12` = Minor of `color {blue} {a_12 = | ( a_21 , a_23 ), ( a_31 ,a_33 ) | } = a_21a_33 - a_23 a_31`

`M_13` = Minor of `color {blue} {a_13 = | ( a_21 , a_22 ), ( a_31 ,a_32 ) | }= a_21a_32 - a_22 a_31`

`M_21` = Minor of `color {blue} {a_21 = | ( a_12 , a_13 ), ( a_32 ,a_33 ) | } = a_12a_33 - a_13 a_32`

`M_22` = Minor of`color {blue} {a_22 = | ( a_11 , a_13 ), ( a_31 ,a_33 ) | } = a_11a_33 - a_13 a_31`
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`M_33` = Minor of `color {blue} {a_33 = | ( a_11 , a_12 ), ( a_21 ,a_22 ) | }= a_11a_22 - a_12 a_21`


Remark : Minor of an element of a determinant of order `n(n ≥ 2)` is a determinant of order `n – 1.`

`color{green} ✍️` Cofactor of an element `a_(ij) `, denoted by `A_(ij)` is defined by

`=>color{red} {"cofactor of element " a_(ij) "is define as " A_(ij) = (-1)^(i+j)M_(ij)}`


`color{red} {A = [ (a_11, a_12 , a_13 ), (a_21, a_22 , a_23 ),( a_31 ,a_32 ,a_33 )] }`

`A_11` = Cofactor of `a_11 = (-1)^(1 +1)M_11 = color {blue} {M_11 = | (a_22, a_23), ( a_32 , a_33 ) |} = a_22 a_33 - a_23 a_32`

`A_12` = Cofactor of `a_12 = (-1)^(1 +2)M_12 = color {blue} { - M_12 = | ( a_21 , a_23 ), ( a_31 ,a_33 ) | } = a_23 a_31 - a_21a_33 `

`A_13` = Cofactor of `a_13 = (-1)^(1 +3)M_13 = color {blue} {M_13 = | ( a_21 , a_22 ), ( a_31 ,a_32 ) | }= a_21a_32 - a_22 a_31`

`A_21` = Cofactor of `a_21 = (-1)^(2 +1)M_21 = color {blue} { - M_21 = | ( a_12 , a_13 ), ( a_32 ,a_33 ) | } =a_13 a_32 - a_12a_33 `
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`A_22` = Cofactor of `a_33 = (-1)^(3 +3)M_33 = color {blue} {M_33 = | ( a_11 , a_12 ), ( a_21 ,a_22 ) | }= a_11a_22 - a_12 a_21`

● Expanding the determinant Δ, along `R_1`, we have

` Δ = (-1)^(1+1) a_11 | (a_22, a_23), ( a_32 , a_33) | + (-1)^(1+2) a_12 | (a_21 , a_23), ( a_31, a_33) | + (-1)^(1+3) a_13 | ( a_21 , a_22 ), ( a_31 , a_32) |`

`= a_11 A_11 + a_12 A_12 + a_13 A_13`, where `A_(ij)` is cofactor of `a_(ij)`

`color{orange}{Δ=" sum of product of elements of " R_1" with their corresponding cofactors"}`

●Hence, Sum of Product of any element with their corresponding Cofactor is equal to the value of Determinants.

`ul color {blue } {text (In Cofactor form ) }`:

`color {green} {Delta = a_11 A_11 + a_12 A_12 + a_13 A_13}`

`color {green} { Delta = a_21 A_21 + a_22 A_22 + a_23 A_23}`

`color {green} { Delta = a_31 A_31 + a_32 A_32 + a_33 A_33}`

`ul color {blue } {text (In Minor form ) }`:

`color{green} {Delta = a_11 M_12 - a_12 M_12 + a_13 M_13}`

`color{green} {Delta = a_21 M_21 - a_22 M_22 + a_23 M_23}`

`color{green} {Delta = a_31 M_31 - a_32 M_32 + a_33 M_33}`



Similarly, Δ can be calculated by other five ways of expansion that is along `C_1, C_2` and `C_3`.

Hence `color{red}{" sum of the product of elements of any row (or column) with their corresponding "}`
`color{red}{"cofactors in a determinant is equal to the value of that determinant "}`

`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