Solution:
So
Now, we will find the matrix for
![\[{{A}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7782def51ee499f2d0c118958f52028e_l3.png "Rendered by QuickLaTeX.com")
we get
Now, we will find the matrix for
![\[\lambda \text{ }A\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d7cb96cb62e8d9a4a037fdced7de038a_l3.png "Rendered by QuickLaTeX.com")
we get
But given,
![\[{{A}^{2}}~=\text{ }\lambda \text{ }A\text{ }+\text{ }\mu \text{ }I\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4744a97391cb6b17d2b0e90f3828864f_l3.png "Rendered by QuickLaTeX.com")
Substitute corresponding values from equation (i) and (ii), we get
And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal
Hence,
![\[\lambda \text{ }+\text{ }0\text{ }=\text{ }4~\Rightarrow ~\lambda \text{ }=\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c012e08f2f1be774b0ebf10f9f8b9ce1_l3.png "Rendered by QuickLaTeX.com")
And also,
![\[2\lambda \text{ }+\text{ }\mu \text{ }=\text{ }7\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-20aa3f61416ee21135baaa033bf7ac77_l3.png "Rendered by QuickLaTeX.com")
Substituting the obtained value of
![\[\lambda \]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1e5278423fe0cd24b56228ebd2dbfcd2_l3.png "Rendered by QuickLaTeX.com")
in the above equation, we get
![\[2\left( 4 \right)\text{ }+\text{ }\mu \text{ }=\text{ }7~\Rightarrow ~8\text{ }+\text{ }\mu \text{ }=\text{ }7~\Rightarrow ~\mu \text{ }=\text{ }\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-652435433592ab322d982dc7e828c4fe_l3.png "Rendered by QuickLaTeX.com")
Therefore, the value of
![\[\lambda \text{ }and\text{ }\mu \text{ }are\text{ }4\text{ }and\text{ }\text{ }1\text{ }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-63e35b63e965c92859f8b508f6171163_l3.png "Rendered by QuickLaTeX.com")
respectively