Given [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], what is [tex]\(\left|A_x\right|\)[/tex]?

A. [tex]\(\left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right|\)[/tex]

B. [tex]\(\left|\begin{array}{ll}5 & 12 \\ 3 & 66\end{array}\right|\)[/tex]

C. [tex]\(\left|\begin{array}{cc}5 & -4 \\ 12 & 66\end{array}\right|\)[/tex]

D. [tex]\(\left|\begin{array}{cc}12 & 66 \\ 3 & 6\end{array}\right|\)[/tex]



Answer :

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].