Question

निम्नलिखित सारणिक के अवयव के उपसारणिक एवं सहखंड ज्ञात कीजिए।

${\displaystyle \left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|}$

Answer 1

उपसारणिक:

${\displaystyle M_{11}=\left|\begin{array}{cc}5& -1\\ 1& 2\end{array}\right|=5\times 2-1=11}$

${\displaystyle M_{12} \left|\begin{array}{cc}3& -1\\ 0& 2\end{array}\right|=3\times 2-0=6}$

${\displaystyle M_{13}=\left|\begin{array}{cc}3& 5\\ 0& 1\end{array}\right|=3\times 1-0=3}$

${\displaystyle M_{21} \left|\begin{array}{cc}0& 4\\ 1& 2\end{array}\right|=0\times 2-1\times 4=-4}$

${\displaystyle M_{22}=\left|\begin{array}{cc}1& 4\\ 0& 2\end{array}\right|=1\times 2-0=2}$

${\displaystyle M_{23}=\left|\begin{array}{cc}1& 0\\ 0& 2\end{array}\right|=1\times 1=1}$

${\displaystyle M_{31}=\left|\begin{array}{cc}0& 4\\ 5& -1\end{array}\right|=-20}$

${\displaystyle M_{32}=\left|\begin{array}{cc}1& 4\\ 3& -1\end{array}\right|=1\times -\left(1\right)-3\times 4=13}$

${\displaystyle M_{33}=\left|\begin{array}{cc}1& 0\\ 3& 5\end{array}\right|=5-0=5}$

सहखंड:

${\displaystyle A_{11}={(-1)}^{1+1} M_{11}=11}$

${\displaystyle A_{12}={(-1)}^{1+2} M_{12}=(-1)\times 6=-6}$

${\displaystyle A_{13}={(-1)}^{1+3} M_{12}=1\cdot 3=3}$

${\displaystyle A_{21}={(-1)}^{2+1}{M}_{21}=(-1)\times (-4)=4}$

${\displaystyle A_{22}={(-1)}^{2+2} M_{22}=1\times 2=2}$

${\displaystyle A_{23}={(-1)}^{2+3} M_{23}=(-1)\cdot 1=-1}$

${\displaystyle A_{31}={(-1)}^{3+1} M_{31}=1\cdot (-20)=-20}$

${\displaystyle A_{32}={(-1)}^{3+2}{M}_{32}=(-1)(-13)=13}$

${\displaystyle A_{33}={(-1)}^{3+3} M_{33}=1\cdot 5=5}$