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# If  $A=\left[\begin{array}{cc}1& 0\\ 2& -1\end{array}\right]$and $B=\left[\begin{array}{cc}3& 1\\ 4& 2\end{array}\right]$ a matrices C such that  $2A+3B+2C$is a null matrix then ___

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a
$\frac{1}{2}\left[\begin{array}{cc}11& 3\\ 8& 4\end{array}\right]$
b
$\frac{1}{2}\left[\begin{array}{cc}-11& -3\\ -16& -8\end{array}\right]$
c
$\frac{1}{2}\left[\begin{array}{cc}11& 3\\ 16& 8\end{array}\right]$
d
$\left[\begin{array}{cc}-11& -3\\ -16& -8\end{array}\right]$

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detailed solution

Correct option is B

$\begin{array}{l}2A+3B+2C=0\\ ⇒2C=-2A-3B\\ =-2\left[\begin{array}{cc}1& 0\\ 2& -1\end{array}\right]-3\left[\begin{array}{cc}3& 1\\ 4& 2\end{array}\right]\\ =\left[\begin{array}{cc}-2-9& 0-3\\ -1-12& 2-6\end{array}\right]\\ =\left[\begin{array}{cc}-11& -3\\ -16& -8\end{array}\right]\\ c=\left[\begin{array}{cc}-\frac{11}{2}& -\frac{3}{2}\\ -8& -4\end{array}\right]\end{array}$