Question

Determine the matrices A and B,where A+2B=[1 2 0,6 -3 3,-5 3 1] and 2A-B=[2 -1 5,2 -1 6,0 1 2]

Answer 1

$\underline{\textsf{Given:}}$

$A+2B=\left(\begin{array}{ccc}1&2&0\\6&-3&3\\-5&3&1\end{array}\right)$

$2A-B=\left(\begin{array}{ccc}2&-1&5\\2&-1&6\\0&1&2\end{array}\right)$

$\underline{\textsf{To find:}}$

$\textsf{The  matrices A and B}$

$\underline{\textsf{Solution:}}$

$\textsf{We apply elimnation method to solve the given matrix equations}$

$\textsf{Multiply (2) by 2 and add it to (1)}$

$A+2B=\left(\begin{array}{ccc}1&2&0\\6&-3&3\\-5&3&1\end{array}\right)$

$4A-2B=\left(\begin{array}{ccc}4&-2&10\\4&-2&12\\0&2&4\end{array}\right)$

$\mathsf{Adding,\;we\;get}$

$5A=\left(\begin{array}{ccc}5&0&10\\10&-5&15\\-5&5&5\end{array}\right)$

$A=\dfrac{1}{5}\left(\begin{array}{ccc}5&0&10\\10&-5&15\\-5&5&5\end{array}\right)$

$\implies\,A=\left(\begin{array}{ccc}1&0&2\\2&-1&3\\-1&1&1\end{array}\right)$

$\mathsf{Put\;A\;in\;(2)}$

$2\left(\begin{array}{ccc}1&0&2\\2&-1&3\\-1&1&1\end{array}\right)-B=\left(\begin{array}{ccc}2&-1&5\\2&-1&6\\0&1&2\end{array}\right)$

$\left(\begin{array}{ccc}2&0&4\\4&-2&6\\-2&2&2\end{array}\right)-B=\left(\begin{array}{ccc}2&-1&5\\2&-1&6\\0&1&2\end{array}\right)$

$B=\left(\begin{array}{ccc}2&0&4\\4&-2&6\\-2&2&2\end{array}\right)-\left(\begin{array}{ccc}2&-1&5\\2&-1&6\\0&1&2\end{array}\right)$

$\implies\boxed{\,B=\left(\begin{array}{ccc}0&1&-1\\2&-1&0\\-2&1&0\end{array}\right)}$

$\underrline{\textsf{Find more:}}$

Find the matix A and B,if 2A+B is equal to (3 -4 2 7) and A-2B is equal to (4 3 1 1)

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