![\left[\begin{array}{cc}a & b \\ c & \frac{1+b c}{a_{7}}\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-22cd42e6a2cb6ad6aecdf53e25617dfb_l3.png "Rendered by QuickLaTeX.com")
![\left[\begin{array}{cc}2 & 5 \\ -3 & 1\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e1505d2b42895090d49723ab4e0e2542_l3.png "Rendered by QuickLaTeX.com")
Solution:
(i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero.
![\left.\begin{array}{l} \text { Now, }|\mathrm{A}|=\frac{\mathrm{a}+\mathrm{abc}}{\mathrm{a}}-\mathrm{bc}=\frac{\mathrm{a}+\mathrm{abc}-\mathrm{abc}}{\mathrm{a}}=1 \neq 0 \\ \text { Therefore, } \mathrm{A}^{-1} \text { exists. } \\ \text { Cofactors of } \mathrm{A} \text { are } \\ C_{11}=\frac{1+\mathrm{bc}}{\mathrm{a}} \\ C_{12}=-\mathrm{C} \\ C_{21}=-\mathrm{b} \\ C_{22}=\mathrm{a} \\ \text { As, } \operatorname{adj} \mathrm{A}=\left[\begin{array}{ll} \mathrm{C}_{11} & \mathrm{C}_{12} \\ \mathrm{C}_{21} & \mathrm{C}_{22} \end{array}\right]^{\mathrm{T}} \\ \mathrm{A}^{\mathrm{T}} \end{array}\right](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6abaf6711a63c2c3ee9cd37a9b2c0327_l3.png "Rendered by QuickLaTeX.com")
(ii) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero.
Now, 
Therefore, 
Cofactors of 
![\text { As, } \operatorname{adj} A=\left[\begin{array}{ll} C_{11} & C_{12} \\ C_{21} & C_{22} \end{array}\right]^{T}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cd13322d348b642dfaeace516a5bb033_l3.png "Rendered by QuickLaTeX.com")
![(\operatorname{adj} A)=\left[\begin{array}{cc} 1 & 3 \\ -5 & 2 \end{array}\right]^{T}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e6c8af93359297e1aea29c382f4da70e_l3.png "Rendered by QuickLaTeX.com")
![=\left[\begin{array}{cc} 1 & -5 \\ 3 & 2 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ef1cf881b0ea573961ec7934d2e6ec79_l3.png "Rendered by QuickLaTeX.com")
![A^{-1}=\frac{1}{17}\left[\begin{array}{cc} 1 & -5 \\ 3 & 2 \end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9bce84b2c79ba94c2126f1c99337d383_l3.png "Rendered by QuickLaTeX.com")
