Answer
See answers below
Work Step by Step
$C_{11}=(-1)^{1+1}.\begin{vmatrix}
1 & 1 &3\\
9 & 0& 2\\
0 & 3 &-1
\end{vmatrix}=1.84=84$
$C_{21}=(-1)^{2+1}.\begin{vmatrix}
0 & 3 &5\\
6 & 0 & 2\\
0& 3 &-1
\end{vmatrix}=(-1).162=-162$
$C_{31}=(-1)^{3+1}.\begin{vmatrix}
0 & 3 &5\\
1 & 1 & 3\\
1 & 3 &-1
\end{vmatrix}=1.18=18$
$C_{41}=(-1)^{4+1}.\begin{vmatrix}
0 & 3 &5\\
1 & 1 & 3\\
9 & 0 &2
\end{vmatrix}=(-1).30=-30$
$C_{12}=(-1)^{1+2}.\begin{vmatrix}
-2 & 1 &3\\
3 & 0& 2\\
2 & 3 &-1
\end{vmatrix}=(-1).1.2=-2$
$C_{22}=(-1)^{2+2}.\begin{vmatrix}
1 & 3 &5\\
3 & 0 & 2\\
2 & 3 &-1
\end{vmatrix}=1.60=60$
$C_{42}=(-1)^{4+2}.\begin{vmatrix}
1 & 3 &5\\
-2 & 1 & 3\\
3 & 0 &2
\end{vmatrix}=1.26=26$
$C_{13}=(-1)^{1+3}.\begin{vmatrix}
-2 & 1 &3\\
3 & 9 & 2\\
2 & 0 &-1
\end{vmatrix}=(-1).29=-29$
$C_{32}=(-1)^{2+3}.\begin{vmatrix}
1 & 3 &5\\
-2 &1 & 3\\
2 & 3 &-1
\end{vmatrix}=(-1).(-38)=38$
$C_{23}=(-1)^{2+3}.\begin{vmatrix}
1 & 0 &5\\
3 & 9 & 2\\
2 & 0 &-1
\end{vmatrix}=(-1).(-99)=99$
$C_{33}=(-1)^{3+3}.\begin{vmatrix}
1 & 0 &5\\
-2 & 1 & 3\\
2 & 0&-1
\end{vmatrix}=(-11).1=-11$
$C_{43}=(-1)^{4+3}.\begin{vmatrix}
1 & 0 &5\\
-2 & 1 & 3\\
2 & 0 &-1
\end{vmatrix}=(-1).(-130)=130$
$C_{14}=(-1)^{4+1}.\begin{vmatrix}
-2 & 1 &1\\
3 & 9 & 0\\
2 & 0 &3
\end{vmatrix}=(-)1.(-81)=81$
$C_{24}=(-1)^{2+4}.\begin{vmatrix}
1 & 0 &3\\
3 & 9 & 0\\
2 & 0 &3
\end{vmatrix}=1.(-27)=-27$
$C_{34}=(-1)^{3+4}.\begin{vmatrix}
1 & 0 &3\\
-2 & 1 & 1\\
2 & 0 &3
\end{vmatrix}=(-1).(-3)=3$
$C_{44}=(-1)^{4+4}.\begin{vmatrix}
1 & 0 &3\\
-2 & 1 & 1\\
3 & 9 &0
\end{vmatrix}=1.(-72)=-72$
a) We have $det(A)=a_{11}C_{11}+a_{21}C_{21}+a_{31}C_{31}+a_{41}C_{41}=-282$
b)$M_C=\begin{bmatrix}
84 &-46 & -29 &81\\
-162& 66&99&-27\\
18 &38 & -11 &3\\
-30 &26 &130 &- 72
\end{bmatrix}$
c) $adj(A)=\begin{bmatrix}
84 &-162 & -18 &-30\\
-46& 60&38&26\\
-29&99 & -11 &130\\
81 &-27 &3 &- 72
\end{bmatrix}$
d) Because $\det(A)=6 \ne 0$, we are able to find $A^{-1}$ by using the theorem: $A^{-1}=\frac{1}{\det(A)}adj(A)=\frac{-1}{282}.\begin{bmatrix}
84 &-162 & -18 &-30\\
-46& 60&38&26\\
-29&99 & -11 &130\\
81 &-27 &3 &- 72
\end{bmatrix}$
