In a third order determinant the factor of a23 is equal to the minor of a23 then the value of the
minor is
Answers
Minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies. e.g. in the determinant D=\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|,D=
∣
∣
∣
∣
∣
∣
∣
a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33
∣
∣
∣
∣
∣
∣
∣
,
minor of {{a}_{12}}a
12
is denoted as {{M}_{12}}=\left| \begin{matrix} {{a}_{21}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{33}} \\ \end{matrix} \right|M
12
=
∣
∣
∣
∣
∣
a
21
a
31
a
23
a
33
∣
∣
∣
∣
∣
and so on.
What are Cofactors?
Cofactor of an element {{a}_{i\,j}}a
ij
is related to its minor as {{C}_{i\,j}}={{\left( -1 \right)}^{i+j}}{{M}_{i\,j}},C
ij
=(−1)
i+j
M
ij
, where ‘i’ denotes the {{i}^{th}}i
th
row and ‘j’ denotes the {{j}^{th}}j
th
column to which the element {{a}_{i\,j}}a
ij
belongs.
Now we define the value of the determinant of order three in terms of ‘Minor’ and ‘Cofactor’ as
D={{a}_{11}}{{M}_{11}}-{{a}_{12}}{{M}_{12}}+{{a}_{13}}{{M}_{13}}\;\;\;\;\; or \;\;\;\;\;D={{a}_{11}}{{C}_{11}}-{{a}_{12}}{{C}_{12}}+{{a}_{13}}{{C}_{13}}D=a
11
M
11
−a
12
M
12
+a
13
M
13
orD=a
11
C
11
−a
12
C
12
+a
13
C
13
Note: (a) A 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.
(b) {{a}_{11}}{{C}_{21}}+{{a}_{12}}{{C}_{22}}+{{a}_{13}}{{C}_{23}}=0,a
11
C
21
+a
12
C
22
+a
13
C
23
=0, i.e. cofactor multiplied to different row/column elements results in zero value.
Row and Column Operations of Determinants
(a) {{R}_{i}}\leftrightarrow {{R}_{j}}\;\;\;or\;\;\; {{C}_{i}}\leftrightarrow {{C}_{j}}, \;\;\;when \;\;\;\;i\ne j;R
i
↔R
j
orC
i
↔C
j
,wheni
=j; This notation is used when we interchange ith row (or column) and jth row (or column).
(b) {{R}_{i}}\leftrightarrow {{C}_{i}};R
i
↔C
i
; This converts the row into the corresponding column.
(c) {{R}_{i}}\to R{{k}_{i}}\;\;\;or\;\;\;{{C}_{i}}\to k{{C}_{i}};\,\,k\in R;R
i
→Rk
i
orC
i
→kC
i
;k∈R; This represents multiplication of ith row (or column) by k.
(d) {{R}_{i}}\to {{R}_{i}}k+{{R}_{j}}\;\;\;\;or\;\;\;Ci\to {{C}_{i}}k+{{C}_{j}};\left( i\ne j \right);R
i
→R
i
k+R
j
orCi→C
i
k+C
j
;(i
=j); This symbol is used to multiply ith row (or column) by k and adding the jth row (or column) to it.