https://file.aiquickdraw.com/mathbot/file/1725536776814nn8ifsgm.jpg

The matrix in the image is given as:

\[
A = \begin{pmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{pmatrix}
\]

You are asked to compute \( A^2 + A + I \), where \( A^2 \) is the square of the matrix, \( A \) is the original matrix, and \( I \) is the identity matrix of the same size.

### Step-by-step Solution:

1. **Matrix \( A \)** is already given:

\[
A = \begin{pmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{pmatrix}
\]

2. **Find \( A^2 \) (matrix multiplication):**

We compute the product of \( A \cdot A \):

\[
A^2 = A \cdot A = \begin{pmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{pmatrix} \cdot \begin{pmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{pmatrix}
\]

We will multiply the matrices now.

\[
A^2 = \begin{pmatrix}
(1 \times 1) + (2 \times 2) + (2 \times 2) & (1 \times 2) + (2 \times 1) + (2 \times 2) & (1 \times 2) + (2 \times 2) + (2 \times 1) \\
(2 \times 1) + (1 \times 2) + (2 \times 2) & (2 \times 2) + (1 \times 1) + (2 \times 2) & (2 \times 2) + (1 \times 2) + (2 \times 1) \\
(2 \times 1) + (2 \times 2) + (1 \times 2) & (2 \times 2) + (2 \times 1) + (1 \times 2) & (2 \times 2) + (2 \times 2) + (1 \times 1)
\end{pmatrix}
\]

Simplifying:

\[
A^2 = \begin{pmatrix}
1 + 4 + 4 & 2 + 2 + 4 & 2 + 4 + 2 \\
2 + 2 + 4 & 4 + 1 + 4 & 4 + 2 + 2 \\
2 + 4 + 2 & 4 + 2 + 2 & 4 + 4 + 1
\end{pmatrix} = \begin{pmatrix}
9 & 8 & 8 \\
8 & 9 & 8 \\
8 & 8 & 9
\end{pmatrix}
\]

So, \( A^2 = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} \).

3. **Identity matrix \( I \)** (since \( A \) is 3x3, we use a 3x3 identity matrix):

\[
I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]

4. **Add \( A^2 \), \( A \), and \( I \):**

\[
A^2 + A + I = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]

First, sum \( A^2 + A \):

\[
A^2 + A = \begin{pmatrix} 9+1 & 8+2 & 8+2 \\ 8+2 & 9+1 & 8+2 \\ 8+2 & 8+2 & 9+1 \end{pmatrix} = \begin{pmatrix} 10 & 10 & 10 \\ 10 & 10 & 10 \\ 10 & 10 & 10 \end{pmatrix}
\]

Then, sum the result with \( I \):

\[
A^2 + A + I = \begin{pmatrix} 10+1 & 10+0 & 10+0 \\ 10+0 & 10+1 & 10+0 \\ 10+0 & 10+0 & 10+1 \end{pmatrix} = \begin{pmatrix} 11 & 10 & 10 \\ 10 & 11 & 10 \\ 10 & 10 & 11 \end{pmatrix}
\]

### Final result:

\[
A^2 + A + I = \begin{pmatrix} 11 & 10 & 10 \\ 10 & 11 & 10 \\ 10 & 10 & 11 \end{pmatrix}
\]

Would you like a further breakdown or have any other questions?

### Relative questions:
1. How is matrix multiplication performed?
2. What is the significance of the identity matrix in matrix algebra?
3. Can we diagonalize the given matrix \( A \)?
4. How do eigenvalues relate to matrix powers?
5. What properties does the resulting matrix exhibit?

### Tip:
When performing matrix operations, double-check each row and column operation to avoid mistakes in addition and multiplication steps.

Find A^2 + A + I for the given matrix A where A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]].

Learn how to compute A^2 + A + I for a 3x3 matrix using matrix multiplication and addition. This problem covers essential matrix operations, including the use of the identity matrix. Suitable for high school students in Grades 10-12.

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