Given Problem:

You have the matrix $A$ as:

$A = \begin{pmatrix} 7 & 4 \\ b & 10 \end{pmatrix}$

where you are instructed to replace $a$ with the last two digits of your BITS ID and $b$ with the month of your birth. You've provided the following replacements:

• Replace $a = 7$ with 28
• Replace $b = 10$ with 10

Thus, the matrix becomes:

$A = \begin{pmatrix} 28 & 4 \\ 10 & 10 \end{pmatrix}$

Steps to Solve:

1. Write the characteristic equation and obtain the eigenvalues.

   The characteristic equation is given by:

   $\text{det}(A - \lambda I) = 0$

   Where $I$ is the identity matrix and $\lambda$ is the eigenvalue.

   For matrix $A$:

   $A - \lambda I = \begin{pmatrix} 28-\lambda & 4 \\ 10 & 10-\lambda \end{pmatrix}$

   The determinant is:

   $\text{det}(A - \lambda I) = (28-\lambda)(10-\lambda) - (4)(10) = 0$

   Expanding this:

   $(28-\lambda)(10-\lambda) - 40 = 0$

   $280 - 28\lambda - 10\lambda + \lambda^2 - 40 = 0$

   $\lambda^2 - 38\lambda + 240 = 0$

   Solving this quadratic equation:

   $\lambda = \frac{38 \pm \sqrt{38^2 - 4(240)}}{2}$

   $\lambda = \frac{38 \pm \sqrt{1444}}{2} = \frac{38 \pm 38}{2}$

   Therefore, the eigenvalues are:

   $\lambda_1 = 38 \quad \text{and} \quad \lambda_2 = 0$

2. Use the Power Method to find the dominant eigenvalue and corresponding eigenvector.

   The dominant eigenvalue is $\lambda_1 = 38$. Using the power method would involve iterating a vector through the matrix $A$ and normalizing it until it converges to the dominant eigenvalue and eigenvector.

   The detailed steps would include starting with an initial vector, multiplying it by $A$, normalizing, and repeating until the vector stabilizes.

3. Under what conditions will the power method fail to converge?

   The power method may fail to converge if the matrix has eigenvalues of the same magnitude but different phases or if the initial vector is orthogonal to the dominant eigenvector.

Questions:

1. What are the general steps to solve a quadratic equation when finding eigenvalues?
2. How can the characteristic polynomial help in determining matrix properties?
3. What are some common issues encountered when using the Power Method for eigenvalue computation?
4. How can the initial vector choice impact the convergence of the Power Method?
5. What are the eigenvectors corresponding to the eigenvalues obtained?

Tip:

When using the Power Method, ensure that the initial vector is not orthogonal to the dominant eigenvector to avoid failure in convergence.