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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants3 Marks
Question
Solve the system of linear equation, using matrix method 5x + 2y = 3; 3x + 2y = 5
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Answer

Matrix form of given equations is AX = B
$\Rightarrow \left[ {\begin{array}{*{20}{c}} 5&2 \\ 3&2 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3 \\ 5 \end{array}} \right]$
Here A = $\left[\begin{array}{ll} 5 & 2 \\ 3 & 2 \end{array}\right]$, X = $\left[\begin{array}{l} x \\ y \end{array}\right]$ and B = $\left[\begin{array}{l} 3 \\ 5 \end{array}\right]$
$\therefore$ |A| = $\left|\begin{array}{ll} 5 & 2 \\ 3 & 2 \end{array}\right|$ = 10 - 6 = 4 $\ne$ 0
Therefore, solution is unique and X = A-1B = $\frac{1}{|A|}$ (adj A) B
$ = \left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} 2&-2 \\ { - 3}&5 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 3 \\ 5 \end{array}} \right]$ 
$= \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {6 - 10} \\ { - 9 + 25} \end{array}} \right] = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} { - 4} \\ {16} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1} \\ 4 \end{array}} \right]$
Therefore, x = -1 and y = 4

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