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Model Paper 5 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 51 Mark
Question
Let $X =\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right], A =\left[\begin{array}{rrr}1 & -1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1\end{array}\right]$ and $B =\left[\begin{array}{l}3 \\ 1 \\ 4\end{array}\right]$. If $A = B$, then $X$ is equal to
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Answer

Given that
$ X=\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right], A=\left[\begin{array}{ccc} 1 & -1 & 2 \\2 & 0 & 1 \\ 3 & 2 & 1 \end{array}\right] \text { and } B=\left[\begin{array}{l} 3 \\ 1 \\ 4 \end{array}\right]$
Also $A X=B$ and we have to find the value of $X$,
Pre$-$multiplying $A ^{-1}$ both sides we gel,
$ A^{-1} AX=A^{-1} B$
$IX=A^{-1} B\left(\because A^{-1} A=I\right)$
$X=A^{-1} B(\because IX=X) ..... (i) $
Now,
$ |A|=\left|\begin{array}{ccc} 1 & -1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1 \end{array}\right|$
$=1(0-2)+1(2-3)+2(4-0)$
$=-2-1+8$
$=5 $
And $\operatorname{adj} A=\left[\begin{array}{ccc}-2 & 5 & -1 \\ 1 & -5 & 3 \\ 4 & -5 & 2\end{array}\right]$
$ \therefore\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right]=\left[\begin{array}{c} -1 \\ 2 \\ 3 \end{array}\right] $
On comparing both sides we get, $x _1=-1, x _2=2$ and $x _3=3$.

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