Suppose you want to know the eigenvalues of the following matrix:

\[ A = \begin{pmatrix} -2 & -24 \\ 3 & 16 \end{pmatrix} \]

1. Create a new document and select Add Calculator.
2. Create the matrix:
   • Press and select Matrix & Vector > Create > Matrix.
   • Enter the number of rows (2) and columns (2). Press and write the matrix values.
3. Store the matrix and name it a: Press and . Write a and press .
4. Compute the eigenvalues:
   • Press and select Matrix & Vector > Advanced > Eigenvalues.
   • Write a inside the brackets of eigVl() and press .

   

   The results should be \( \lambda_1 = 4 \) and \( \lambda_2 = 10 \), which are the eigenvalues.

Consider the following matrix:

\[ A = \begin{pmatrix} -2 & -24 \\ 3 & 16 \end{pmatrix} \]

Suppose you want to compute the eigenvectors associated with \( \lambda_1 = 4 \) and \( \lambda_2 = 10 \).

1. Write the homogeneous linear system associated with the eigenvalue: \[ \left( A - \lambda I \right) \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] Here: \[ \begin{cases} -6x - 24y = 0 \\ 3x + 12y = 0 \end{cases} \]
2. Solve the system using the calculator:
   • Press and select Algebra > Solve System of Equations > Solve System of Linear Equations.
   • The result should display \( y \) as a free variable and \( x = -4y \).

   

3. Interpret the solution:

   The eigenvectors are represented as: \[ t \begin{pmatrix} -4 \\ 1 \end{pmatrix} \] Here, \( t \) is a free variable. A possible eigenvector is: \[ x_1 = \begin{pmatrix} -4 \\ 1 \end{pmatrix} \]

4. Repeat the process for \( \lambda_2 = 10 \). A possible eigenvector is: \[ x_2 = \begin{pmatrix} -2 \\ 1 \end{pmatrix} \]