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To find the inverse of the given 2x2 matrix, we will follow these steps:

The matrix given is:

\[
\begin{pmatrix}
2 & 2 \\
2 & 7
\end{pmatrix}
\]

### Step 1: Calculate the determinant.

The formula for the determinant of a 2x2 matrix:

\[
\text{det}(A) = ad - bc
\]

where the matrix is

\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]

For your matrix:

\[
a = 2, \quad b = 2, \quad c = 2, \quad d = 7
\]

\[
\text{det}(A) = (2)(7) - (2)(2) = 14 - 4 = 10
\]

### Step 2: Check if the determinant is zero.

Since the determinant is 10 (which is not zero), the matrix is **invertible**.

### Step 3: Use the inverse formula for a 2x2 matrix.

The inverse of a 2x2 matrix is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

Substituting the values:

\[
A^{-1} = \frac{1}{10} \begin{pmatrix} 7 & -2 \\ -2 & 2 \end{pmatrix}
\]

Thus, the inverse is:

\[
A^{-1} = \begin{pmatrix} \frac{7}{10} & -\frac{2}{10} \\ -\frac{2}{10} & \frac{2}{10} \end{pmatrix}
\]

or simplified:

\[
A^{-1} = \begin{pmatrix} 0.7 & -0.2 \\ -0.2 & 0.2 \end{pmatrix}
\]

### Final Answer:
- Option **A** is correct. The inverse matrix is:

\[
\begin{pmatrix} 0.7 & -0.2 \\ -0.2 & 0.2 \end{pmatrix}
\]

---

Would you like further details or have any questions?

Here are five related questions to explore:

1. How do we compute the inverse of larger matrices (3x3 or more)?
2. What happens if the determinant of a matrix is zero?
3. Can every square matrix be inverted?
4. What are the practical applications of matrix inverses?
5. How does matrix multiplication work with inverse matrices?

**Tip:** If a matrix has a zero determinant, it is called "singular" and does not have an inverse.

Find the inverse of the given matrix: [ [2, 2], [2, 7] ].

Learn how to find the inverse of a 2x2 matrix: [2, 2; 2, 7]. This problem covers key linear algebra concepts, including matrix inversion and the determinant. Suitable for high school students in Grades 10-12.

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