If A = [3 1 4, 0 2 6, 0 0 5] 
 then eigenvalues of A*A + 2A + I 
 are 

Given the matrix \( A = \begin{bmatrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \end{bmatrix} \), we are tasked with finding the eigenvalues of the matrix \( A^TA + 2A + I \).

### Step 1: Calculate \( A^TA \)
First, we compute the transpose of \( A \), denoted \( A^T \), and then find \( A^TA \).

Matrix \( A \):
\[
A = \begin{bmatrix}
3 & 1 & 4 \\
0 & 2 & 6 \\
0 & 0 & 5
\end{bmatrix}
\]

Transpose of \( A \) is:
\[
A^T = \begin{bmatrix}
3 & 0 & 0 \\
1 & 2 & 0 \\
4 & 6 & 5
\end{bmatrix}
\]

Now, multiply \( A^T \) by \( A \):
\[
A^TA = \begin{bmatrix}
3 & 0 & 0 \\
1 & 2 & 0 \\
4 & 6 & 5
\end{bmatrix}
\begin{bmatrix}
3 & 1 & 4 \\
0 & 2 & 6 \\
0 & 0 & 5
\end{bmatrix}
\]

Performing the multiplication:
\[
A^TA = \begin{bmatrix}
9 & 3 & 12 \\
3 & 5 & 16 \\
12 & 16 & 57
\end{bmatrix}
\]

### Step 2: Calculate \( 2A \)
We now calculate \( 2A \) by multiplying each element of \( A \) by 2:
\[
2A = \begin{bmatrix}
6 & 2 & 8 \\
0 & 4 & 12 \\
0 & 0 & 10
\end{bmatrix}
\]

### Step 3: Calculate \( A^TA + 2A + I \)
We now add the three matrices \( A^TA \), \( 2A \), and the identity matrix \( I \), which is:
\[
I = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]

First, sum \( A^TA \) and \( 2A \):
\[
A^TA + 2A = \begin{bmatrix}
9 & 3 & 12 \\
3 & 5 & 16 \\
12 & 16 & 57
\end{bmatrix} +
\begin{bmatrix}
6 & 2 & 8 \\
0 & 4 & 12 \\
0 & 0 & 10
\end{bmatrix}
=
\begin{bmatrix}
15 & 5 & 20 \\
3 & 9 & 28 \\
12 & 16 & 67
\end{bmatrix}
\]

Now, add the identity matrix \( I \):
\[
A^TA + 2A + I = \begin{bmatrix}
15 & 5 & 20 \\
3 & 9 & 28 \\
12 & 16 & 67
\end{bmatrix} + 
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} =
\begin{bmatrix}
16 & 5 & 20 \\
3 & 10 & 28 \\
12 & 16 & 68
\end{bmatrix}
\]

### Step 4: Find Eigenvalues
Now, we need to find the eigenvalues of the matrix:
\[
\begin{bmatrix}
16 & 5 & 20 \\
3 & 10 & 28 \\
12 & 16 & 68
\end{bmatrix}
\]

Let me compute the eigenvalues of this matrix.The eigenvalues of the matrix \( A^TA + 2A + I \) are approximately:

\[
\lambda_1 \approx 96.55, \quad \lambda_2 \approx 12.69, \quad \lambda_3 \approx 4.76
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we calculate the determinant of a matrix?
2. What is the significance of eigenvectors corresponding to these eigenvalues?
3. How can we verify if a matrix is diagonalizable?
4. What role do eigenvalues play in solving systems of differential equations?
5. How is the trace of a matrix related to its eigenvalues?

**Tip:** The sum of the eigenvalues of a matrix is equal to the trace (sum of diagonal elements) of the matrix.

If A = [3 1 4, 0 2 6, 0 0 5] then eigenvalues of A*A + 2A + I are

Learn how to find the eigenvalues of the matrix A*A + 2A + I, where A is a 3x3 matrix. This problem covers key concepts in linear algebra, including matrix multiplication, matrix transpose, and the computation of eigenvalues. Suitable for undergraduate students in mathematics.

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