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Q. Find the cofactors of elements $a_{12}, a_{22}, a_{32}$, respectively of the matrix $\begin{bmatrix}1&sin\,\theta&1\\ -sin\,\theta&1&sin\,\theta\\ -1&-sin\,\theta&1\end{bmatrix}$

Determinants

Solution:

$M_{12}=\begin{vmatrix}-sin\,\theta&sin\,\theta\\ -1&1\end{vmatrix}=-sin\,\theta+sin\,\theta=0$
$\Rightarrow c_{12}=-M_{12}=0$
$M_{22}=\begin{vmatrix}1&1\\ -1&1\end{vmatrix}=1+1=2$
$\Rightarrow C_{22}=M_{22}=2$
$M_{32}=\begin{vmatrix}1&1\\ -sin\,\theta&sin\,\theta\end{vmatrix}=sin\,\theta+sin\,\theta=2\,sin\,\theta$
$c_{32}=-M_{32}=-2sin\,\theta$