We are given a matrix equation and need to find the determinant of a specific matrix [tex]\( \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right| \)[/tex].
Given the system of equations represented by the matrix equation:
[tex]\[
\left[\begin{array}{cc}5 & -4 \\ 3 & 6\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}12 \\ 66\end{array}\right]
\][/tex]
We need to first confirm that we are dealing with the correct matrix [tex]\( \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right| \)[/tex].
Let's find its determinant:
[tex]\[
\left| \begin{array}{cc} 12 & -4 \\ 66 & 6 \end{array} \right|
\][/tex]
To calculate the determinant of a [tex]\(2 \times 2\)[/tex] matrix [tex]\( \left|\begin{array}{cc}a & b \\ c & d\end{array}\right| \)[/tex], we use the formula:
[tex]\[
ad - bc
\][/tex]
Now apply this formula:
[tex]\[
\left| \begin{array}{cc} 12 & -4 \\ 66 & 6 \end{array} \right| = (12 \cdot 6) - (-4 \cdot 66)
\][/tex]
Calculate each term:
[tex]\[
12 \cdot 6 = 72
\][/tex]
[tex]\[
-4 \cdot 66 = -264
\][/tex]
Combine these results:
[tex]\[
72 - (-264) = 72 + 264 = 336
\][/tex]
Thus, the determinant of the matrix [tex]\( \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right| \)[/tex] is:
[tex]\[
336
\][/tex]
So, the determinant [tex]\(\left|A_x\right| = 336\)[/tex].
