To find the inverse of a 2x2 matrix, we can apply the formula for the inverse of a matrix [tex]\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\)[/tex], which is given by:
[tex]\[
\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]^{-1} = \frac{1}{ad - bc} \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]
\][/tex]
Given the matrix:
[tex]\[
\left[\begin{array}{cc} 30 & 8 \\ 6 & 2 \end{array}\right]
\][/tex]
Let's denote:
[tex]\( a = 30 \)[/tex],
[tex]\( b = 8 \)[/tex],
[tex]\( c = 6 \)[/tex], and
[tex]\( d = 2 \)[/tex].
First, we need to calculate the determinant [tex]\(ad - bc\)[/tex]:
[tex]\[
ad - bc = (30 \cdot 2) - (8 \cdot 6) = 60 - 48 = 12
\][/tex]
Since the determinant is not zero, the matrix is invertible.
Next, we apply the inverse formula:
[tex]\[
\left[\begin{array}{cc} 30 & 8 \\ 6 & 2 \end{array}\right]^{-1} = \frac{1}{12} \left[\begin{array}{cc} 2 & -8 \\ -6 & 30 \end{array}\right]
\][/tex]
Multiplying each element of the matrix [tex]\(\left[\begin{array}{cc} 2 & -8 \\ -6 & 30 \end{array}\right]\)[/tex] by [tex]\(\frac{1}{12}\)[/tex]:
[tex]\[
\left[\begin{array}{cc} \frac{2}{12} & \frac{-8}{12} \\ \frac{-6}{12} & \frac{30}{12} \end{array}\right] = \left[\begin{array}{cc} \frac{1}{6} & -\frac{2}{3} \\ -\frac{1}{2} & \frac{5}{2} \end{array}\right]
\][/tex]
Now, we need to convert these fractions to decimals and round to the nearest hundredth:
[tex]\[
\left[\begin{array}{cc} \frac{1}{6} & -\frac{2}{3} \\ -\frac{1}{2} & \frac{5}{2} \end{array}\right] \approx \left[\begin{array}{cc} 0.17 & -0.67 \\ -0.50 & 2.50 \end{array}\right]
\][/tex]
Therefore, the inverse of the matrix [tex]\(\left[\begin{array}{cc} 30 & 8 \\ 6 & 2 \end{array}\right]\)[/tex] is:
[tex]\[
\left[\begin{array}{cc} 0.17 & -0.67 \\ -0.50 & 2.50 \end{array}\right]
\][/tex]
