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Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
Question
If $\begin{bmatrix}\text{x}&4&1\end{bmatrix}\begin{bmatrix}2&1&2\\1&0&2\\0&2&-4\end{bmatrix}\begin{bmatrix}\text{x}\\4\\-1\end{bmatrix}=0,$ find x.
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Answer

Given,
$\begin{bmatrix}\text{x}&4&1\end{bmatrix}\begin{bmatrix}2&1&2\\1&0&2\\0&2&-4\end{bmatrix}\begin{bmatrix}\text{x}\\4\\-1\end{bmatrix}=0$
$\Rightarrow\begin{bmatrix}2\text{x}+4+0&\text{x}+0+2&2\text{x}+8-4\end{bmatrix}\begin{bmatrix}\text{x}\\4\\-1\end{bmatrix}=0$
$\Rightarrow\begin{bmatrix}2\text{x}+4&\text{x}+2&2\text{x}+4\end{bmatrix}\begin{bmatrix}\text{x}\\4\\-1\end{bmatrix}=0$
$\Rightarrow\begin{bmatrix}(2\text{x}+4)\text{x}+4(\text{x}+2)-1(2\text{x}+4)\end{bmatrix}=0$
$\Rightarrow2\text{x}^2+4\text{x}+4\text{x}+8-2\text{x}-4=0$
$\Rightarrow2\text{x}+6\text{x}+4=0$
$\Rightarrow2\text{x}^2+2\text{x}+4\text{x}+4=0$
$\Rightarrow2\text{x}(\text{x}+1)+4(\text{x}+1)=0$
$\Rightarrow(\text{x}+1)+(2\text{x}+4)=0$
$\Rightarrow\text{x}+1=0\ \text{or }2\text{x}+4=0$
$\Rightarrow\text{x}=-1\ \text{or }\text{x}=-2$
Hence, x = -1 or -2

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