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The problem involves two tasks:

1. **Finding all values of \( \alpha \) that make the matrix singular** using the LU decomposition method (Doolittle’s method).
2. **Finding the smallest positive integer value of \( \alpha \)** to solve the system \( A\mathbf{x} = [1, 2, 3]^T \) using Doolittle’s method.

The matrix \( A \) is:

\[
A = \begin{pmatrix}
1 & 2 & \alpha \\
\alpha & 8 & 2\alpha \\
2\alpha & 2\alpha & 9
\end{pmatrix}
\]

### Task 1: Find values of \( \alpha \) that make the matrix singular
A matrix is singular if its determinant is zero. Let's compute the determinant of matrix \( A \) and solve for \( \alpha \).

\[
\text{det}(A) = \text{det} \begin{pmatrix}
1 & 2 & \alpha \\
\alpha & 8 & 2\alpha \\
2\alpha & 2\alpha & 9
\end{pmatrix}
\]

We set the determinant to zero and solve for \( \alpha \).

### Task 2: Solve \( A \mathbf{x} = [1, 2, 3]^T \)
To find the solution using **Doolittle’s method**, the matrix \( A \) will be decomposed into the form \( A = LU \), where \( L \) is a lower triangular matrix and \( U \) is an upper triangular matrix. We will use the smallest positive integer value of \( \alpha \) from Task 1 to solve for \( \mathbf{x} \).

Let me first compute the determinant and check for singular values of \( \alpha \).The determinant of the matrix \( A \) is:

\[
\text{det}(A) = 2\alpha^3 - 12\alpha^2 - 18\alpha + 72
\]

To find the values of \( \alpha \) that make the matrix singular, we solve for \( \alpha \) when the determinant is set to zero:

\[
2\alpha^3 - 12\alpha^2 - 18\alpha + 72 = 0
\]

Let me solve this cubic equation for \( \alpha \).The solutions for \( \alpha \) involve complex expressions due to the nature of the cubic equation, suggesting non-integer or complex roots. For practical purposes, it may be helpful to approximate the real roots numerically.

Let me compute the approximate values of \( \alpha \).It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, from the form of the determinant, the next step is to either factorize it or use numerical methods to find approximate values of \( \alpha \). Once the smallest positive real value of \( \alpha \) is found, it can be used in the second part of the problem to solve the system using Doolittle's method.

Do you want to proceed with a different approach, or would you like more details on Doolittle's method?

Here are some related questions to explore:
1. What is the process for LU decomposition using Doolittle's method?
2. How does a matrix become singular in general?
3. Can cubic equations like this always be solved analytically?
4. What is the significance of the determinant in matrix analysis?
5. How does numerical approximation help in solving higher-degree equations?

**Tip:** When solving cubic equations, numerical methods like Newton's method or built-in solvers can be helpful for practical computations.

Find all values of alpha that make the matrix singular using the LU decomposition method by Doolittle's method. Then use the smallest positive integer value of alpha to solve the system Ax = [1, 2, 3]^T using Doolittle's method.

Learn how to find the values of alpha that make a matrix singular and use Doolittle’s method for LU decomposition to solve the system Ax = [1, 2, 3]^T. This detailed explanation covers key concepts in matrix algebra, including singular matrices and LU decomposition.

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