Answer :

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.