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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
Solve system of linear equations, using matrix method.
5x + 2y = 4
7x + 3y = 5
✓

Answer

Matrix form of given equations is AX = B $\Rightarrow\ \begin{bmatrix}5&2\\7&3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}4\\5\end{bmatrix}$
$\text{Here}\ \text{A}=\begin{bmatrix}5&2\\7&3\end{bmatrix},\ \text{X}=\begin{bmatrix}x\\y\end{bmatrix}\text{and B}=\begin{bmatrix}4\\5\end{bmatrix}$
$\therefore \ \text{|A|}=\begin{vmatrix}5&2\\7&5\end{vmatrix}=15-14=1\neq0$
Therefore, solution is unique and $\text{X=A}^{-1}\text{B}=\frac{1}{\text{|A|}}\text{(adj. A)B}$
$\Rightarrow\ \begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{1}\begin{bmatrix}3&-2\\-7&5\end{bmatrix}\begin{bmatrix}4\\5\end{bmatrix}=\begin{bmatrix}12-10\\-28+25\end{bmatrix}=\begin{bmatrix}2\\-3\end{bmatrix}$
Therefore, x = 2 and y = -3

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