To find the inverse of a 2x2 matrix, we begin with a given matrix:
[tex]\[ A =
\begin{pmatrix}
14 & 1 \\
16 & 1 \\
\end{pmatrix}
\][/tex]
The formula for finding the inverse of a 2x2 matrix
[tex]\[ A =
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
\][/tex]
is given by:
[tex]\[ A^{-1} = \frac{1}{ad - bc}
\begin{pmatrix}
d & -b \\
-c & a \\
\end{pmatrix}
\][/tex]
Here, we identify [tex]\(a = 14\)[/tex], [tex]\(b = 1\)[/tex], [tex]\(c = 16\)[/tex], and [tex]\(d = 1\)[/tex]. We need to calculate the determinant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = ad - bc = (14 \times 1) - (16 \times 1) = 14 - 16 = -2 \][/tex]
Since the determinant [tex]\(\Delta \neq 0\)[/tex], the matrix has an inverse.
Now, substituting [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] along with [tex]\(\Delta\)[/tex] into the inverse formula, we get:
[tex]\[ A^{-1} = \frac{1}{-2}
\begin{pmatrix}
1 & -1 \\
-16 & 14 \\
\end{pmatrix}
=
\begin{pmatrix}
\frac{1}{-2} & \frac{-1}{-2} \\
\frac{-16}{-2} & \frac{14}{-2} \\
\end{pmatrix}
=
\begin{pmatrix}
-0.5 & 0.5 \\
8 & -7 \\
\end{pmatrix}
\][/tex]
Thus, the inverse of the given matrix
[tex]\[ A =
\begin{pmatrix}
14 & 1 \\
16 & 1 \\
\end{pmatrix}
\][/tex]
is
[tex]\[ A^{-1} =
\begin{pmatrix}
-0.5 & 0.5 \\
8 & -7 \\
\end{pmatrix}
\][/tex]
This result is already rounded to the nearest hundredth. Therefore, the inverse of the matrix is:
[tex]\[
\begin{pmatrix}
-0.5 & 0.5 \\
8 & -7 \\
\end{pmatrix}
\][/tex]
