Question

Solve for the matrices A and B, when 2A + B = $\left[\begin{array}{ccc}3&-4\\2&7\end{array}\right] and \hspace{3} A- 2B = \left[\begin{array}{ccc}4&3\\1&1\end{array}\right]$

Answer 1

➡️Solution:

it is given that

$2A+B=\left[\begin{array}{ccc}3&-4\\2&7\end{array}\right]\:\:\:eq1\\\\A-2B=\left[\begin{array}{ccc}4&3\\1&1\end{array}\right]\:\:\:eq2$
multiply eq1 by 2 and add eq1 and eq2

$4A+2B=\left[\begin{array}{ccc}6&-8\\4&14\end{array}\right]\\\\4A+2B+A-2B=\left[\begin{array}{ccc}6&-8\\4&14\end{array}\right]+\left[\begin{array}{ccc}4&3\\1&1\end{array}\right]$

$5A=\left[\begin{array}{ccc}6+4&-8+3\\4+1&14+1\end{array}\right]\\\\5A=\left[\begin{array}{ccc}10&-5\\5&15\end{array}\right]\\\\A=\left[\begin{array}{ccc}2&-1\\1&3\end{array}\right]$

to find the value of B matrix,put the value of A matrix in eq1 or in eq2

$2\left[\begin{array}{ccc}2&-1\\1&3\end{array}\right]+B=\left[\begin{array}{ccc}3&-4\\2&7\end{array}\right]\\\\ \left[\begin{array}{ccc}4&-2\\2&6\end{array}\right]+B=\left[\begin{array}{ccc}3&-4\\2&7\end{array}\right]\\\\B=\left[\begin{array}{ccc}3&-4\\2&7\end{array}\right]-\left[\begin{array}{ccc}4&-2\\2&6\end{array}\right]\\\\B=\left[\begin{array}{ccc}3-4&-4+2\\2-2&7-6\end{array}\right]\\\\B=\left[\begin{array}{ccc}-1&-2\\0&1\end{array}\right]$
Hope it helps you.