Sure, let's find the inverse of the given matrix step-by-step, and also round the elements of the inverse to the nearest hundredth.
The original matrix is:
[tex]\[
\mathbf{A} = \begin{pmatrix}
4 & 3 \\
2 & 6
\end{pmatrix}
\][/tex]
To find the inverse of this 2x2 matrix, we can use the formula for the inverse of a 2x2 matrix:
[tex]\[
\mathbf{A}^{-1} = \frac{1}{\text{det}(\mathbf{A})} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\][/tex]
where [tex]\(\mathbf{A} = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\)[/tex], and [tex]\(\text{det}(\mathbf{A}) = ad - bc\)[/tex].
### Step-by-Step Calculation
1. Calculate the determinant of [tex]\(\mathbf{A}\)[/tex]:
[tex]\[
\text{det}(\mathbf{A}) = (4)(6) - (3)(2) = 24 - 6 = 18
\][/tex]
2. Construct the adjugate matrix:
[tex]\[
\text{Adj}(\mathbf{A}) = \begin{pmatrix}
6 & -3 \\
-2 & 4
\end{pmatrix}
\][/tex]
3. Divide the adjugate matrix by the determinant to get the inverse:
[tex]\[
\mathbf{A}^{-1} = \frac{1}{18} \begin{pmatrix}
6 & -3 \\
-2 & 4
\end{pmatrix} = \begin{pmatrix}
\frac{6}{18} & \frac{-3}{18} \\
\frac{-2}{18} & \frac{4}{18}
\end{pmatrix} = \begin{pmatrix}
0.33333333 & -0.16666667 \\
-0.11111111 & 0.22222222
\end{pmatrix}
\][/tex]
4. Round the elements of the inverse matrix to the nearest hundredth:
[tex]\[
\mathbf{A}^{-1} \approx \begin{pmatrix}
0.33 & -0.17 \\
-0.11 & 0.22
\end{pmatrix}
\][/tex]
Therefore, the inverse of the matrix
[tex]\[
\begin{pmatrix}
4 & 3 \\
2 & 6
\end{pmatrix}
\][/tex]
is
[tex]\[
\begin{pmatrix}
0.33 & -0.17 \\
-0.11 & 0.22
\end{pmatrix}
\][/tex]
So, filling in the missing values, we have:
[tex]\[
\begin{pmatrix}
4 & 3 \\
2 & 6
\end{pmatrix} \quad \rightarrow \quad
\begin{pmatrix}
0.33 & -0.17 \\
-0.11 & 0.22
\end{pmatrix}
\][/tex]
