To find the inverse of the given matrix, we start with the matrix:
[tex]\[
\begin{bmatrix}
9 & 2 \\
16 & 4
\end{bmatrix}
\][/tex]
The formula to find the inverse of a 2x2 matrix
[tex]\[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\][/tex]
is given by:
[tex]\[
\frac{1}{ad - bc} \begin{bmatrix}
d & -b \\
-c & a
\end{bmatrix}
\][/tex]
Let's apply this formula to our matrix.
First, identify the elements of the matrix:
[tex]\[
a = 9, \quad b = 2, \quad c = 16, \quad d = 4
\][/tex]
Next, we need to calculate the determinant [tex]\( ad - bc \)[/tex]:
[tex]\[
\text{Determinant} = ad - bc = (9 \times 4) - (2 \times 16) = 36 - 32 = 4
\][/tex]
Since the determinant is non-zero (in this case, 4), the matrix has an inverse.
The inverse matrix [tex]\( \frac{1}{ad - bc} \begin{bmatrix}
d & -b \\
-c & a
\end{bmatrix}
\)[/tex] in our case is:
[tex]\[
\frac{1}{4} \begin{bmatrix}
4 & -2 \\
-16 & 9
\end{bmatrix}
\][/tex]
Now, multiply each element of the matrix by [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[
\begin{bmatrix}
4 \times \frac{1}{4} & -2 \times \frac{1}{4} \\
-16 \times \frac{1}{4} & 9 \times \frac{1}{4}
\end{bmatrix}
=
\begin{bmatrix}
1 & -0.5 \\
-4 & 2.25
\end{bmatrix}
\][/tex]
So, the inverse of the matrix
[tex]\[
\begin{bmatrix}
9 & 2 \\
16 & 4
\end{bmatrix}
\][/tex]
is
[tex]\[
\begin{bmatrix}
1 & -0.5 \\
-4 & 2.25
\end{bmatrix}
\][/tex]
All the elements are already rounded as needed. Therefore, the inverse matrix rounded to the nearest hundredth is:
[tex]\[
\begin{bmatrix}
1 & -0.5 \\
-4 & 2.25
\end{bmatrix}
\][/tex]
