If A = | cc 3 & -2 -1 & 4 |, B = | c 2 a 1 |, C = | c -4 5 |, D =… - Vidyadip

Question

If $A =\left|\begin{array}{cc}3 & -2 \\ -1 & 4\end{array}\right|, B =\left|\begin{array}{c}2 a \\ 1\end{array}\right|, C =\left|\begin{array}{c}-4 \\ 5\end{array}\right|, D =\left|\begin{array}{l}2 \\ b \end{array}\right|$ and $AB +2 C =4 D$ then find the values of $a$ and $b$.

✓

Answer

$\begin{array}{l}A=\left|\begin{array}{cc}3 & -2 \\ -1 & 4\end{array}\right|_{2 \times 2}, B=\left|\begin{array}{c}2 a \\ 1\end{array}\right|_{2 \times 1}\end{array} $
$ C =\left|\begin{array}{c}-4 \\ 5\end{array}\right|_{2 \times 1}, D=\left|\begin{array}{l}2 \\ b \end{array}\right|_{2 \times 1} $
$ AB =\left|\begin{array}{cc}3 & -2 \\ -1 & 4\end{array}\right|\left|\begin{array}{c}2 a \\ 1\end{array}\right| $
$ =\left|\begin{array}{c}6 a -2 \\ -2 a +4\end{array}\right|_{2 \times 1} $
$ 2 C =\left|\begin{array}{c}-8 \\ 10\end{array}\right| $
$ AB +2 C =\left|\begin{array}{c}6 a -2 \\ -2 a +4\end{array}\right|\left|\begin{array}{c}-8 \\ 10\end{array}\right| $
$ =\left|\begin{array}{c}6 a -10 \\ -2 a +14\end{array}\right|_{2 \times 1} $
$ 4 D =\left|\begin{array}{c}8 \\ 4 b \end{array}\right|$
Given, $A B+2 C=4 D$
$ \begin{array}{l} \left|\begin{array}{c} 6 a-10 \\ -2 a+14 \end{array}\right|=\left|\begin{array}{c} 8 \\ 4 b \end{array}\right| \\\end{array} $
$6 a-10=8$
$\Rightarrow 6 a=18$
$\Rightarrow a=3$
$-2 a+14=4 b$
$\Rightarrow-2(3)+14=4 b$
$\Rightarrow 8=4 b$
$\Rightarrow 2=b$

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