To find the inverse of the matrix
[tex]\[
\begin{pmatrix}
2 & 0 \\
9 & 1
\end{pmatrix},
\][/tex]
we follow these steps:
### Step 1: Determine the determinant
The determinant of the matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is calculated as
[tex]\[
\text{det} = ad - bc.
\][/tex]
For our specific matrix:
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} =
\begin{pmatrix}
2 & 0 \\
9 & 1
\end{pmatrix},
\][/tex]
the determinant is:
[tex]\[
\text{det} = (2)(1) - (0)(9) = 2.
\][/tex]
### Step 2: Check if the determinant is non-zero
Since the determinant (2) is non-zero, the matrix is invertible.
### Step 3: Find the adjugate of the matrix
The adjugate (or adjoint) of the matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is computed by swapping the elements on the main diagonal and changing the sign of the off-diagonal elements:
[tex]\[
\text{adj}
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} =
\begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}.
\][/tex]
Applying this to our matrix:
[tex]\[
\begin{pmatrix}
2 & 0 \\
9 & 1
\end{pmatrix}
,
\][/tex]
the adjugate matrix is:
[tex]\[
\begin{pmatrix}
1 & -0 \\
-9 & 2
\end{pmatrix} =
\begin{pmatrix}
1 & 0 \\
-9 & 2
\end{pmatrix}.
\][/tex]
### Step 4: Calculate the inverse
The inverse of a matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is given by
[tex]\[
\text{inverse} = \frac{1}{\text{det}} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}.
\][/tex]
For our matrix, this becomes:
[tex]\[
\text{inverse} = \frac{1}{2} \begin{pmatrix}
1 & 0 \\
-9 & 2
\end{pmatrix} = \begin{pmatrix}
\frac{1}{2} & 0 \\
-\frac{9}{2} & 1
\end{pmatrix} = \begin{pmatrix}
0.5 & 0 \\
-4.5 & 1
\end{pmatrix}.
\][/tex]
Thus, the inverse of the matrix
[tex]\[
\begin{pmatrix}
2 & 0 \\
9 & 1
\end{pmatrix}
\][/tex]
is
[tex]\[
\begin{pmatrix}
0.5 & 0 \\
-4.5 & 1
\end{pmatrix}.
\][/tex]
### Step 5: Rounding (if necessary)
In this case, the matrix elements are already precise to the hundredth place, so no additional rounding is necessary.
Therefore, the inverse of the given matrix is
[tex]\[
\begin{pmatrix}
0.5 & 0 \\
-4.5 & 1
\end{pmatrix}.
\][/tex]
