![X+Y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-feae1467403cbe6d2f22b4ff13bcf8d1_l3.png "Rendered by QuickLaTeX.com")
![X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-83fad44b696a719ac577a807f6f301c3_l3.png "Rendered by QuickLaTeX.com")
![2 X+3 Y=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-44d62e9232d9a9706d3aec20dcaa6cb0_l3.png "Rendered by QuickLaTeX.com")
![3 X+2 Y=\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9d3aedf62c5dbd0ecb18f67717d6c05f_l3.png "Rendered by QuickLaTeX.com")
(i) Adding both the expressions we get,
![(X+Y)+(X-Y)=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]+\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ba5fb484383d2f15ece17b4b8335fb1c_l3.png "Rendered by QuickLaTeX.com")
![2 X=\left[\begin{array}{ll}10 & 0 \\ 2 & 8\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fc5b59b1bd8c73dca15feb36fad03331_l3.png "Rendered by QuickLaTeX.com")
![\mathrm{X}=\frac{1}{2}\left[\begin{array}{ll}10 & 0 \\ 2 & 8\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 1 & 4\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4ad8ed8ec934039de86ce8fe4e7d1df8_l3.png "Rendered by QuickLaTeX.com")
Subtracting both the expressions we get,
![(X+Y)-(X-Y)=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4ce51d3c0e5875a6f34a3ff5a6cc0208_l3.png "Rendered by QuickLaTeX.com")
![2 \mathrm{Y}=\left[\begin{array}{ll}4 & 0 \\ 2 & 2\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cef4e08ae6d84d9ac81c35c1d28bbb52_l3.png "Rendered by QuickLaTeX.com")
![\mathrm{Y}=\frac{1}{2}\left[\begin{array}{ll}4 & 0 \\ 2 & 2\end{array}\right]=\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b13e53d6b320c518a0a4a05fc26fcf9b_l3.png "Rendered by QuickLaTeX.com")
(ii) ![2 X+3 Y=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]_{\ldots . .(1)}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1122b60b14ffcd00a6d3f3e6a9fda612_l3.png "Rendered by QuickLaTeX.com")
![3 X+2 Y=\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right] \ldots](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bfbc9517e7c68f53a94b19fa317fa847_l3.png "Rendered by QuickLaTeX.com")
Multiplying equation (1) by 2 we get,
![4 X+6 Y=2\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]=\left[\begin{array}{ll}4 & 6 \\ 8 & 0\end{array}\right]_{\ldots \ldots .}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1e539dd99cef98a2c3566192882cbccb_l3.png "Rendered by QuickLaTeX.com")
Multiplying equation (2) by 3 we get,
![9 X+6 Y=3\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right]=\left[\begin{array}{rc}6 & -6 \\ -3 & 15\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c9d0c1ac2cd9d1a180826e808db3d2b0_l3.png "Rendered by QuickLaTeX.com")
Subtracting equation (4) from (3) we get,
![-5 X=\left[\begin{array}{ll}4 & 6 \\ 8 & 0\end{array}\right]-\left[\begin{array}{cc}6 & -6 \\ -3 & 15\end{array}\right]=\left[\begin{array}{cc}-2 & 12 \\ 11 & -15\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6696625fda5d5a01ec841823afee3f55_l3.png "Rendered by QuickLaTeX.com")
![X=\frac{-1}{5}\left[\begin{array}{cc}-2 & 12 \\ 11 & -15\end{array}\right]=\left[\begin{array}{cc}\frac{2}{5} & \frac{-12}{5} \\ \frac{-11}{5} & 3\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a8dff4f165b9ef88527f8dcab7ce8ff7_l3.png "Rendered by QuickLaTeX.com")
since We know that
![2 \mathrm{X}+3 \mathrm{Y}=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0640a7af87c3b3eae7cc4288795fdf30_l3.png "Rendered by QuickLaTeX.com")
Substituting the value of 
![2\left[\begin{array}{ll}\frac{2}{5} & \frac{-12}{5} \\ \frac{-1}{5} & 3\end{array}\right]+3 Y=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1c1102a830b0e0cab1acb83aadc10063_l3.png "Rendered by QuickLaTeX.com")
![\left[\begin{array}{lr}\frac{4}{5} & -\frac{24}{5} \\ -\frac{22}{5} & 6\end{array}\right]+3 Y=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3a6fb9696a45ed59bb0507b13038c894_l3.png "Rendered by QuickLaTeX.com")
![3 \mathrm{Y}=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]-\left[\begin{array}{ll}\frac{4}{5} & -\frac{24}{5} \\ -\frac{22}{5} & 6\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-57bfde8a68ae23acabb4369cdefb8eba_l3.png "Rendered by QuickLaTeX.com")
therefore,
![3 Y=\left[\begin{array}{cc}2-\frac{4}{5} & 3+\frac{24}{5} \\ 4+\frac{22}{5} & 0-6\end{array}\right]=\left[\begin{array}{ll}\frac{6}{5} & \frac{39}{5} \\ \frac{42}{5} & -6\end{array}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ae76550136d3ad5e0d3953160a09c403_l3.png "Rendered by QuickLaTeX.com")
