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The given problem involves a matrix \( A \) and the task is to find the expression \( A^2 + 5A + 7I \), where \( I \) is the identity matrix.

### Matrix \( A \):
\[
A = \begin{pmatrix} 
3 & 1 & 4 \\ 
2 & 1 & 3 \\ 
2 & 2 & 1 
\end{pmatrix}
\]

### Steps to solve \( A^2 + 5A + 7I \):

1. **Compute \( A^2 \) (matrix multiplication of \( A \) with itself):**

\[
A^2 = A \times A
\]

2. **Multiply \( A \) by 5:**

\[
5A = 5 \times A
\]

3. **Multiply the identity matrix \( I \) by 7:**

\[
7I = 7 \times I
\]

where \( I \) is the 3x3 identity matrix:
\[
I = \begin{pmatrix} 
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 
\end{pmatrix}
\]

4. **Add the resulting matrices from steps 1, 2, and 3 to get the final result:**

\[
A^2 + 5A + 7I
\]

Let me calculate this for you.The result of the expression \( A^2 + 5A + 7I \) is:

\[
A^2 + 5A + 7I = \begin{pmatrix} 
41 & 17 & 39 \\ 
24 & 21 & 29 \\ 
22 & 16 & 27 
\end{pmatrix}
\]

Would you like a detailed explanation of the steps? Here are some related questions to further explore:

1. How do you perform matrix multiplication?
2. What are the properties of identity matrices in operations like this?
3. Can matrix powers be applied to non-square matrices?
4. How do scalar multiplications of matrices affect their elements?
5. How does matrix addition work element-wise?

**Tip:** When multiplying matrices, ensure the number of columns in the first matrix matches the number of rows in the second for a valid multiplication.

Find A^2 + 5A + 7I for the matrix A = [[3, 1, 4], [2, 1, 3], [2, 2, 1]].

Learn how to solve the expression A^2 + 5A + 7I for a given 3x3 matrix. This problem involves matrix multiplication, scalar multiplication, and the use of the identity matrix. Suitable for undergraduate-level students studying linear algebra.

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