Question

$Give \: A = \left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right] \:, B = \left[ \begin{array}{cc}-3 & - 2 \\ 4 & 0 \end{array} \right] \: , C = \left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]$
$Find \: \: the \: \: matrix \: \: x \: \: such \: \: that \: \: \\ A \: + 2x = 2B + C$
​

Answer 1

Given :-

• $\sf\: A = \left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]$
• $\sf\:B=\left[ \begin{array}{cc}-3 & - 2 \\ 4 & 0 \end{array} \right]$
• $\sf\:C = \left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]$

To find :-

• Matrix x.

Solution :-

A + 2x = 2B + C

$:\implies\sf{\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]+2x=2\left[ \begin{array}{cc}-3 & - 2 \\ 4 & 0 \end{array} \right]+\left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]}$

$:\implies\sf{\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]+2x=\left[ \begin{array}{cc}-6 & - 4 \\ 8 & 0 \end{array} \right]+\left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]}$

$:\implies\sf{2x=\left[ \begin{array}{cc}-6 & - 4 \\ 8 & 0 \end{array} \right]+\left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]-\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]}$

$:\implies\sf{2x=\left[\begin{array}{cc}(-6-4-2)&(-4+0+6)\\(8+0-2)&(0+2-0)\end{array}\right]}$

$:\implies\sf{2x=\left[\begin{array}{cc}12&2\\6&2\end{array}\right]}$

$:\implies\sf{x=\dfrac{1}{2}\left[\begin{array}{cc}12&2\\6&2\end{array}\right]}$

$:\implies\sf{x=\left[\begin{array}{cc}6&1\\3&1\end{array}\right]}$

Answer 2

$\bf{Given \::\:} \begin{cases} A = \left[ \begin{array}{cc}2 & - 6 \\\\ 2 & 0 \end{array} \right] \:\\\\ B = \left[ \begin{array}{cc}-3 & - 2 \\\\ 4 & 0 \end{array} \right] \: \\\\ C = \left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]\end{cases}$

Need To Find : $\sf{\: \: The \: \: matrix \: \: x \: \: such \: \: that \: \: A \: + 2x = 2B + C}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\qquad \qquad \underline {\bf{ \: \: A \: + 2x = 2B + C}}$

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}$

$:\implies\sf{\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]+2x=2\left[ \begin{array}{cc}-3 & - 2 \\ 4 & 0 \end{array} \right]+\left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]}$

$:\implies\sf{\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]+2x=\left[ \begin{array}{cc}-6 & - 4 \\ 8 & 0 \end{array} \right]+\left[ \begin{array}{cc}-4 & 0 \\ 0 & 2 \end{array} \right]}$

$:\implies\sf{\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]+2x=\left[ \begin{array}{cc}-6+(-4) & - 4+0 \\ 8+0 & 0+2 \end{array} \right]}$

$:\implies\sf{2x= \left[ \begin{array}{cc}-6+(-4) & - 4+0 \\ 8+0 & 0+2 \end{array} \right]-\left[ \begin{array}{cc}2 & - 6 \\ 2 & 0 \end{array} \right]}$

$\begin{gathered}:\implies\sf{2x=\left[\begin{array}{cc}(-6-4-2)&(-4+0+6)\\(8+0-2)&(0+2-0)\end{array}\right]}\end{gathered}$

$:\implies\sf{2x=\left[\begin{array}{cc}12&2\\6&2\end{array}\right]}$

$:\implies\sf{x=\dfrac{1}{2}\left[\begin{array}{cc}12&2\\6&2\end{array}\right]}$

⠀⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {x=\left[\begin{array}{cc}6&1\\3&1\end{array}\right]}}}}\:\bf{\bigstar}$

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