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To determine the determinant of the matrix [tex]\( A_x \)[/tex], we need to solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] using the given system of linear equations:
[tex]\[ \left[\begin{array}{cc} 5 & -4 \\ 3 & 6 \end{array}\right] \left[\begin{array}{c} x \\ y \end{array}\right] = \left[\begin{array}{c} 12 \\ 66 \end{array}\right] \][/tex]
Given that [tex]\(\left|A_x\right| = \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right|\)[/tex], we calculate the determinant of the matrix [tex]\(\left[\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right]\)[/tex].
The determinant of a 2x2 matrix [tex]\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\)[/tex] is given by:
[tex]\[ \text{det} = (a \cdot d) - (b \cdot c) \][/tex]
Using the entries of the given matrix [tex]\(\left[\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right]\)[/tex]:
[tex]\[ a = 12, \quad b = -4, \quad c = 66, \quad d = 6 \][/tex]
Plugging these values into the determinant formula:
[tex]\[ \text{det} = (12 \cdot 6) - (-4 \cdot 66) \][/tex]
[tex]\[ \text{det} = 72 + 264 \][/tex]
[tex]\[ \text{det} = 336 \][/tex]
Therefore, the determinant [tex]\( \left|A_x\right| \)[/tex] is:
[tex]\[ \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right| = 336 \][/tex]
[tex]\[ \left[\begin{array}{cc} 5 & -4 \\ 3 & 6 \end{array}\right] \left[\begin{array}{c} x \\ y \end{array}\right] = \left[\begin{array}{c} 12 \\ 66 \end{array}\right] \][/tex]
Given that [tex]\(\left|A_x\right| = \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right|\)[/tex], we calculate the determinant of the matrix [tex]\(\left[\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right]\)[/tex].
The determinant of a 2x2 matrix [tex]\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\)[/tex] is given by:
[tex]\[ \text{det} = (a \cdot d) - (b \cdot c) \][/tex]
Using the entries of the given matrix [tex]\(\left[\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right]\)[/tex]:
[tex]\[ a = 12, \quad b = -4, \quad c = 66, \quad d = 6 \][/tex]
Plugging these values into the determinant formula:
[tex]\[ \text{det} = (12 \cdot 6) - (-4 \cdot 66) \][/tex]
[tex]\[ \text{det} = 72 + 264 \][/tex]
[tex]\[ \text{det} = 336 \][/tex]
Therefore, the determinant [tex]\( \left|A_x\right| \)[/tex] is:
[tex]\[ \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right| = 336 \][/tex]
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