Answer
(a) Expansion by cofactors of the second row, we have
$$
\left|\begin{array}{rrr}{-3} & {2} & {2} \\ {4} & {5} & {6} \\ {2} & {-3} & {1}\end{array}\right|
=4C_{21}+5C_{22}+6C_{23}=4(-5)+5(-5)+6(-5)=-20-25-30=-75.
$$
(b) Expansion by cofactors of the second column, we have
$$
\left|\begin{array}{rrr}{-3} & {2} & {2} \\ {4} & {5} & {6} \\ {2} & {-3} & {1}\end{array}\right|
=2C_{12}+5C_{22}-3C_{32}=2(8)+5(-5)-3(22)=16-25-66=-75.
$$
Work Step by Step
Given
$$
\left|\begin{array}{rrr}{-3} & {2} & {1} \\ {4} & {5} & {6} \\ {2} & {-3} & {1}\end{array}\right|.
$$
The cofactors, are given by:
\begin{align*}
C_{11}&=23, \quad C_{12}=8, \quad C_{13}=-22,\\
C_{21}&=-5, \quad C_{22}=-5, \quad C_{23}=-5,\\
C_{31}&=7, \quad C_{32}=22, \quad C_{33}=-23.
\end{align*}
(a) Expansion by cofactors of the second row, we have
$$
\left|\begin{array}{rrr}{-3} & {2} & {2} \\ {4} & {5} & {6} \\ {2} & {-3} & {1}\end{array}\right|
=4C_{21}+5C_{22}+6C_{23}=4(-5)+5(-5)+6(-5)=-20-25-30=-75.
$$
(b) Expansion by cofactors of the second column, we have
$$
\left|\begin{array}{rrr}{-3} & {2} & {2} \\ {4} & {5} & {6} \\ {2} & {-3} & {1}\end{array}\right|
=2C_{12}+5C_{22}-3C_{32}=2(8)+5(-5)-3(22)=16-25-66=-75.
$$
