To find the inverse of the matrix [tex]\( A \)[/tex], we proceed as follows:
Given the matrix:
[tex]\[
A = \begin{pmatrix}
6 & 1 \\
11 & 2
\end{pmatrix}
\][/tex]
The formula for 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]
We'll apply this formula step by step:
1. Identify the elements of the matrix:
[tex]\[
a = 6, \quad b = 1, \quad c = 11, \quad d = 2
\][/tex]
2. Calculate the determinant ([tex]\(\Delta\)[/tex]) of [tex]\( A \)[/tex], which is:
[tex]\[
\Delta = ad - bc = (6 \cdot 2) - (1 \cdot 11) = 12 - 11 = 1
\][/tex]
3. Formulate the matrix substituting the values accordingly:
[tex]\[
A^{-1} = \frac{1}{1} \begin{pmatrix}
2 & -1 \\
-11 & 6
\end{pmatrix}
= \begin{pmatrix}
2 & -1 \\
-11 & 6
\end{pmatrix}
\][/tex]
Thus, the inverse of matrix [tex]\( A \)[/tex] is:
[tex]\[
A^{-1} = \begin{pmatrix}
2 & -1 \\
-11 & 6
\end{pmatrix}
\][/tex]
To ensure clarity and precision, particularly for sensitive computational applications, the final values for the elements of the inverse matrix could be written in great detail:
[tex]\[
A^{-1} \approx \begin{pmatrix}
2.0000000000000018 & -1.0000000000000009 \\
-11.00000000000001 & 6.000000000000005
\end{pmatrix}
\][/tex]
These values demonstrate the exactitude required in a mathematical context.
