The inverse of the matrix \(\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\) is

To find the inverse of the matrix

\(A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\)

We can use the formula for the inverse of a \( 2 \times 2 \) matrix:

\(A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

where \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).

Identify \( a, b, c, d \)}

In our matrix, we have:

\(a = 2, \quad b = 1, \quad c = 1, \quad d = 1\)

Calculate the determinant \( ad - bc \)

Calculating the determinant:

\(ad - bc = (2)(1) - (1)(1) = 2 - 1 = 1\)

Calculate the inverse:

Now substituting into the formula for the inverse:

\(A^{-1} = \frac{1}{1} \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}\)

Thus, the inverse of the matrix \( \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \) is \(\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}\)

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