`color{green} ✍️` If `A = [(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)]` be a matrix of order `3 × 3, `
then the determinant of `A` is defined as `color{orange} |A| = |(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)| `
`color{red}{"Expansion of matrix "`
`color{green} ✍️` There are six ways of expanding a determinant of order 3 corresponding to each of three rows `(R_1 , R_2 and R_3)` and three columns `(C_1, C_2 and C3)` giving the same value as shown below
`|A| = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|`
Expansion along first Row `(R_1)`
`color {green} text{Step 1 }` Multiply first element `a_(11) `of `R_1` by `(-1)^(1+1)` and with the second order determinant obtained by deleting the elements of first row `(R_1)` and first column `(C_1)` of `| A |` as `a_(11)` lies in `R_1` and `C_1`
i.e., `color {blue} {(-1)^(1+2) a_12 | (a_22, a_23),(a_32 ,a_33) |`
`color {green} text {Step 2 }` Multiply 2nd element `a_12` of `R_1` by `(–1)^(1 + 2)` `[(–1)^text(sum of suffixes in) a_12]` and the second order determinant obtained by deleting elements of first row `(R_1)` and 2nd column `(C_2)` of | A | as `a_12` lies in `R_1` and `C_2`,
i.e., `color {blue} {(-1)^(1 +2) a _12 | (a_21 , a _23),(a_31, a_33) |`
`color {green} text{ Step 3 }` Multiply third element `a_13` of `R_1` by `(–1)^(1 + 3)` `[(–1)^text(sum of suffixes in) a_13] ` and the second order determinant obtained by deleting elements of first row `(R_1)` and third column `(C_3)` of | A | as `a_13` lies in `R_1` and `C_3`,
i.e, `color {blue} {(-1)^(1+3) a_13 | (a_21, a_22), (a_31 ,a_32) |`
`color {green} text { Step 4 }` Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps 1, 2 and 3 above is given by
`det A = |A| = (–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 ) |`
or ` |A| = a_11 (a_22 ,a_33 – a_32 a_23) – a_12 (a_21 a_33 – a_31 a_23) + a_13 (a_21 a_32 – a_31 a_22)`
`color {blue} { = a_11 a_22 a_33 – a_11 a_32 a_23 – a_12 a_21 a_33 + a_12 a_31 a_23 + a_13 a_21 a_32
– a_13 a_31 a_22}` .....................(1)
`color{green} ✍️` If `A = [(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)]` be a matrix of order `3 × 3, `
then the determinant of `A` is defined as `color{orange} |A| = |(a_(11), a_(12),a_13),(a_(21), a_(22),a_23),(a_31,a_32,a_33)| `
`color{red}{"Expansion of matrix "`
`color{green} ✍️` There are six ways of expanding a determinant of order 3 corresponding to each of three rows `(R_1 , R_2 and R_3)` and three columns `(C_1, C_2 and C3)` giving the same value as shown below
`|A| = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|`
Expansion along first Row `(R_1)`
`color {green} text{Step 1 }` Multiply first element `a_(11) `of `R_1` by `(-1)^(1+1)` and with the second order determinant obtained by deleting the elements of first row `(R_1)` and first column `(C_1)` of `| A |` as `a_(11)` lies in `R_1` and `C_1`
i.e., `color {blue} {(-1)^(1+2) a_12 | (a_22, a_23),(a_32 ,a_33) |`
`color {green} text {Step 2 }` Multiply 2nd element `a_12` of `R_1` by `(–1)^(1 + 2)` `[(–1)^text(sum of suffixes in) a_12]` and the second order determinant obtained by deleting elements of first row `(R_1)` and 2nd column `(C_2)` of | A | as `a_12` lies in `R_1` and `C_2`,
i.e., `color {blue} {(-1)^(1 +2) a _12 | (a_21 , a _23),(a_31, a_33) |`
`color {green} text{ Step 3 }` Multiply third element `a_13` of `R_1` by `(–1)^(1 + 3)` `[(–1)^text(sum of suffixes in) a_13] ` and the second order determinant obtained by deleting elements of first row `(R_1)` and third column `(C_3)` of | A | as `a_13` lies in `R_1` and `C_3`,
i.e, `color {blue} {(-1)^(1+3) a_13 | (a_21, a_22), (a_31 ,a_32) |`
`color {green} text { Step 4 }` Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps 1, 2 and 3 above is given by
`det A = |A| = (–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 ) |`
or ` |A| = a_11 (a_22 ,a_33 – a_32 a_23) – a_12 (a_21 a_33 – a_31 a_23) + a_13 (a_21 a_32 – a_31 a_22)`
`color {blue} { = a_11 a_22 a_33 – a_11 a_32 a_23 – a_12 a_21 a_33 + a_12 a_31 a_23 + a_13 a_21 a_32
– a_13 a_31 a_22}` .....................(1)