Question

if the value of a 3*3 determinant is 3, then the value of the determinant formed by its co factors will be-
a)9
b)3
c)27
d)none of these

Answer 1

Task. If the value of a $3*3$ determinant is 3, then the value of the determinant formed by its cofactors will be

a) 9

b) 3

c) 27

d) none of these

Solution. Let

$A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\cr a_{21}&a_{22}&a_{23}\cr a_{31}&a_{32}&a_{33}\end{pmatrix}$

Recall that $(i,j)$-minor $M_{ij}$ is the matrix obtained from $A$ by removing $i$-th row and $j$-th column. Then the $(i,j)$-cofactor is defined as

$C_{ij}=(-1)^{i+j}\det(M_{ij}).$

For example, if $i=1$ and $j=3$, then

$M_{13}=\begin{pmatrix}a_{21}&a_{22}\cr a_{31}&a_{32}\end{pmatrix}$

and

$C_{13}=(-1)^{1+3}\det(M_{13})=+\begin{vmatrix}a_{21}&a_{22}\cr a_{31}&a_{32}\end{vmatrix}=a_{21}a_{32}-a_{31}a_{22}.$

Let

$C=\begin{pmatrix}C_{11}&C_{21}&C_{31}\cr C_{12}&C_{22}&C_{32}\cr C_{13}&C_{23}&C_{33}\end{pmatrix}$

be the matrix consisting of cofactors and transposed.

Then it is known that

$AC=\begin{pmatrix}\det(A)&0&0\cr 0&\det(A)&0\cr 0&0&\det(A)\end{pmatrix}$

Hence

$\det(A)\det(C)=\det(AC)=\det(A)^{3}.$

Therefore

$\det(C)=\det(A)^{3}/\det(A)=\det(A)^{2}=3^{2}=9.$

Answer. a) $\det(C)=9$