How Can I Determine The Minimal Polynomial Of A Matrix A With Distinct Eigenvalues Λ1 And Λ2, But With A Non-trivial Jordan Block Structure, Specifically When A Is A 6x6 Matrix And The Jordan Block Corresponding To Λ1 Is 3x3 And The Jordan Block Corresponding To Λ2 Is 3x3 As Well?

To determine the minimal polynomial of a matrix ${ A }$ which is a 6x6 matrix with distinct eigenvalues ${ \lambda_1 }$ and ${ \lambda_2 }$, each having a 3x3 Jordan block, we need to consider the structure of the Jordan blocks.

1. Eigenvalues and Jordan Blocks: The matrix ${ A }$ has two distinct eigenvalues ${ \lambda_1 }$ and ${ \lambda_2 }$, each with a single 3x3 Jordan block. This means the largest Jordan block for each eigenvalue is 3x3.

2. Minimal Polynomial Factors: For each eigenvalue, the minimal polynomial will have a factor of ${ (x - \lambda)^3 }$ because the largest Jordan block size is 3.

3. Combining Factors: Since ${ \lambda_1 }$ and ${ \lambda_2 }$ are distinct, the minimal polynomial is the product of the minimal polynomials for each eigenvalue. Therefore, the minimal polynomial will be the product of ${ (x - \lambda_1)^3 }$ and ${ (x - \lambda_2)^3 }$.

4. Conclusion: The minimal polynomial of matrix ${ A }$ is the product of these factors, which is ${ (x - \lambda_1)^3 (x - \lambda_2)^3 }$.

Thus, the minimal polynomial of matrix ${ A }$ is ${\boxed{(x - \lambda_1)^3 (x - \lambda_2)^3}}$.