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We are given the matrix \( A \) as:

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

We are tasked with finding the value of \( A^2 - A + I \), where \( I \) is the identity matrix.

Let's proceed step by step:

### Step 1: Compute \( A^2 \)
We need to calculate the square of matrix \( A \), which means multiplying \( A \) by itself.

\[
A^2 = A \times A = \begin{bmatrix} 
1 & 2 & 2 \\
1 & 2 & 2 \\
2 & 2 & 1 
\end{bmatrix} \times \begin{bmatrix} 
1 & 2 & 2 \\
1 & 2 & 2 \\
2 & 2 & 1 
\end{bmatrix}
\]

To compute \( A^2 \), perform matrix multiplication for each element:

- First row, first column: \( 1(1) + 2(1) + 2(2) = 1 + 2 + 4 = 7 \)
- First row, second column: \( 1(2) + 2(2) + 2(2) = 2 + 4 + 4 = 10 \)
- First row, third column: \( 1(2) + 2(2) + 2(1) = 2 + 4 + 2 = 8 \)

- Second row, first column: \( 1(1) + 2(1) + 2(2) = 1 + 2 + 4 = 7 \)
- Second row, second column: \( 1(2) + 2(2) + 2(2) = 2 + 4 + 4 = 10 \)
- Second row, third column: \( 1(2) + 2(2) + 2(1) = 2 + 4 + 2 = 8 \)

- Third row, first column: \( 2(1) + 2(1) + 1(2) = 2 + 2 + 2 = 6 \)
- Third row, second column: \( 2(2) + 2(2) + 1(2) = 4 + 4 + 2 = 10 \)
- Third row, third column: \( 2(2) + 2(2) + 1(1) = 4 + 4 + 1 = 9 \)

Thus, we have:

\[
A^2 = \begin{bmatrix} 
7 & 10 & 8 \\
7 & 10 & 8 \\
6 & 10 & 9 
\end{bmatrix}
\]

### Step 2: Compute \( A^2 - A \)
Now subtract matrix \( A \) from matrix \( A^2 \):

\[
A^2 - A = \begin{bmatrix} 
7 & 10 & 8 \\
7 & 10 & 8 \\
6 & 10 & 9 
\end{bmatrix} - \begin{bmatrix} 
1 & 2 & 2 \\
1 & 2 & 2 \\
2 & 2 & 1 
\end{bmatrix}
\]

Perform the element-wise subtraction:

\[
A^2 - A = \begin{bmatrix} 
7 - 1 & 10 - 2 & 8 - 2 \\
7 - 1 & 10 - 2 & 8 - 2 \\
6 - 2 & 10 - 2 & 9 - 1 
\end{bmatrix} = \begin{bmatrix} 
6 & 8 & 6 \\
6 & 8 & 6 \\
4 & 8 & 8 
\end{bmatrix}
\]

### Step 3: Add the identity matrix \( I \)
Finally, we add the identity matrix \( I \), which is:

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

Now, compute \( A^2 - A + I \):

\[
A^2 - A + I = \begin{bmatrix} 
6 & 8 & 6 \\
6 & 8 & 6 \\
4 & 8 & 8 
\end{bmatrix} + \begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\]

Perform the element-wise addition:

\[
A^2 - A + I = \begin{bmatrix} 
6 + 1 & 8 + 0 & 6 + 0 \\
6 + 0 & 8 + 1 & 6 + 0 \\
4 + 0 & 8 + 0 & 8 + 1 
\end{bmatrix} = \begin{bmatrix} 
7 & 8 & 6 \\
6 & 9 & 6 \\
4 & 8 & 9 
\end{bmatrix}
\]

Thus, the value of \( A^2 - A + I \) is:

\[
\boxed{\begin{bmatrix} 
7 & 8 & 6 \\
6 & 9 & 6 \\
4 & 8 & 9 
\end{bmatrix}}
\]

Would you like further details or clarifications?

Here are 5 related questions:
1. How do you determine whether a matrix is invertible?
2. What properties do identity matrices have?
3. How is matrix multiplication defined in terms of dimensions?
4. How do you find the determinant of a 3x3 matrix?
5. What is the trace of a matrix, and what is its significance?

**Tip:** When multiplying matrices, always check the dimensions to ensure that the number of columns in the first matrix matches the number of rows in the second matrix.

If A = [[1, 2, 2], [1, 2, 2], [2, 2, 1]], find the value of A^2 - A + I.

Solve the matrix problem where A = [[1, 2, 2], [1, 2, 2], [2, 2, 1]] and you are tasked with finding A^2 - A + I. Learn about matrix multiplication, addition, and how the identity matrix affects results. Suitable for high school and early college students.

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