To find the inverse of the matrix:
[tex]\[
\begin{pmatrix}
3 & 9 \\
7 & 18
\end{pmatrix}
\][/tex]
we need to proceed through the following steps:
### Step 1: Calculate the Determinant
The determinant of a [tex]\(2 \times 2\)[/tex] matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is given by the formula:
[tex]\[
\text{det}(A) = ad - bc
\][/tex]
So for our matrix:
[tex]\[
\begin{pmatrix}
3 & 9 \\
7 & 18
\end{pmatrix}
\][/tex]
the determinant is:
[tex]\[
\text{det}(A) = (3 \cdot 18) - (9 \cdot 7) = 54 - 63 = -9
\][/tex]
### Step 2: Check if the Determinant is Zero
If the determinant is zero, the matrix is not invertible. In this case, the determinant is [tex]\(-9\)[/tex], which is not zero, so the matrix is invertible.
### Step 3: Find the Inverse Matrix
The inverse of a [tex]\(2 \times 2\)[/tex] matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is given by:
[tex]\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] from our matrix:
[tex]\[
\begin{pmatrix}
3 & 9 \\
7 & 18
\end{pmatrix}
\][/tex]
Thus,
[tex]\[
A^{-1} = \frac{1}{-9} \begin{pmatrix}
18 & -9 \\
-7 & 3
\end{pmatrix}
= \begin{pmatrix}
\frac{18}{-9} & \frac{-9}{-9} \\
\frac{-7}{-9} & \frac{3}{-9}
\end{pmatrix}
= \begin{pmatrix}
-2 & 1 \\
0.77777778 & -0.33333333
\end{pmatrix}
\][/tex]
### Step 4: Round the Elements of the Inverse Matrix
Finally, we round the elements of the inverse matrix to the nearest hundredth:
[tex]\[
\begin{pmatrix}
-2 & 1 \\
0.77777778 & -0.33333333
\end{pmatrix}
\][/tex]
After rounding, we get:
[tex]\[
\begin{pmatrix}
-2.00 & 1.00 \\
0.78 & -0.33
\end{pmatrix}
\][/tex]
### Conclusion
The inverse of the matrix
[tex]\[
\begin{pmatrix}
3 & 9 \\
7 & 18
\end{pmatrix}
\][/tex]
is:
[tex]\[
\begin{pmatrix}
-2.00 & 1.00 \\
0.78 & -0.33
\end{pmatrix}
\][/tex]
