Question

यदि A^-1 = ${\displaystyle \left[\begin{array}{ccc}3& -1& 1\\ -15& 6& -5\\ 5& -2& 2\end{array}\right]}$ और B = ${\displaystyle \left[\begin{array}{ccc}1& 2& -2\\ -1& 3& 0\\ 0& -2& 1\end{array}\right]}$, हो तो (AB)^-1 का मान ज्ञात कीजिए।

Answer 1

हम जानते हैं कि, ${\displaystyle {\left(AB\right)}^{-1}=B^{-1}{A}^{-1}}$.

B = ${\displaystyle \left[\begin{array}{ccc}1& 2& -2\\ -1& 3& 0\\ 0& -2& 1\end{array}\right]}$

∴ |B| = ${\displaystyle 1\times 3-2\times (-1)-2\left(2\right)}$

= 3 + 2 - 4

= 5 - 4

= 1

अब,

${\displaystyle A_{11}=3,A_{12}=1,A_{13}=2}$

${\displaystyle A_{21}=2,A_{22}=1,A_{23}=2}$

${\displaystyle A_{31}=6,A_{32}=2,A_{33}=5}$

∴ adjB = ${\displaystyle \left[\begin{array}{ccc}3& 2& 6\\ 1& 1& 2\\ 2& 2& 5\end{array}\right]}$

अब,

${\displaystyle B^{-1}=\frac{1}{\left|B\right|}.adjB}$

∴ ${\displaystyle B^{-1}=\left[\begin{array}{ccc}3& 2& 6\\ 1& 1& 2\\ 2& 2& 5\end{array}\right]}$

∴ ${\displaystyle {\left(AB\right)}^{-1}=B^{-1}{A}^{-1}}$

= ${\displaystyle \left[\begin{array}{ccc}3& 2& 6\\ 1& 1& 2\\ 2& 2& 5\end{array}\right]\left[\begin{array}{ccc}3& -1& 1\\ -15& 6& -5\\ 5& -2& 2\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}9-30+30& -3+12-12& 3-10+12\\ 3-15+10& -1+6-4& 1-5+4\\ 6-30+25& -2+12-10& 2-10+10\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}9& -3& 5\\ -2& 1& 0\\ 1& 0& 2\end{array}\right]}$