Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
Solve system of linear equations, using matrix method.
2x + y + z = 1
x - 2y - z = $\frac{3}{2}$
3y - 5z = 9
✓
Answer
Matrix from of given equations is AX = B $\Rightarrow\ \begin{bmatrix}2&1&1\\1&-2&-1\\0&3&-5\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\\frac{3}{2}\\9\end{bmatrix}$
$\text{Here}\ \text{A}=\begin{bmatrix}2&1&1\\1&-2&-1\\0&3&-5\end{bmatrix},\ \text{X}=\begin{bmatrix}x\\y\\z\end{bmatrix}\text{and B}=\begin{bmatrix}1\\\frac{3}{2}\\9\end{bmatrix}$
$\therefore\ \text{|A|}= \begin{bmatrix}2&1&1\\1&-2&-1\\0&3&-5\end{bmatrix}=2(10+3)-1(-5-0)+1(3-0)=26+5+3=34\neq0$
Therefore, solution is unique and $\text{X=A}^{-1}\text{B}=\frac{1}{\text{|A|}}\text{(adj. A)B}$
$\Rightarrow\ \begin{bmatrix}x\\y\\z\end{bmatrix}=\frac{1}{34}\begin{bmatrix}13&8&1\\5&-10&3\\3&-6&-5\end{bmatrix}\begin{bmatrix}1\\\frac{3}{2}\\9\end{bmatrix}$
$=\frac{1}{34}\begin{bmatrix}13+12+9\\5-15+27\\3-9-45\end{bmatrix}=\frac{1}{34}\begin{bmatrix}34\\17\\-51\end{bmatrix}=\begin{bmatrix}1\\\frac{1}{2}\\\frac{-3}{2}\end{bmatrix}$
Therefore, $x=1,y=\frac{1}{2}\text{and}\ z=\frac{3}{2}$
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