Consider the following matrix A=[23xy]. If the eigen values of A are 4 an 8, then

Given the matrix \( A = \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix} \), and the eigenvalues of \( A \) are \( \lambda_1 = 4 \) and \( \lambda_2 = 8 \), we can find the values of \( x \) and \( y \).

### Steps to Solve:

1. **Eigenvalue Equation**:
   The eigenvalues \( \lambda \) of a matrix \( A \) satisfy the characteristic equation:
   \[
   \text{det}(A - \lambda I) = 0
   \]
   where \( I \) is the identity matrix of the same dimension as \( A \).

2. **Form the Characteristic Polynomial**:
   For the matrix \( A = \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix} \), the characteristic polynomial is:
   \[
   \text{det} \left( \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) = \text{det} \begin{pmatrix} 2-\lambda & 3 \\ x & y-\lambda \end{pmatrix}
   \]
   The determinant of this matrix is:
   \[
   (2-\lambda)(y-\lambda) - 3x
   \]
   Expanding this:
   \[
   \lambda^2 - (2+y)\lambda + (2y - 3x) = 0
   \]

3. **Use the Given Eigenvalues**:
   The roots of the characteristic polynomial \( \lambda^2 - (2+y)\lambda + (2y - 3x) = 0 \) are given as 4 and 8.

   So, the characteristic polynomial can also be written as:
   \[
   (\lambda - 4)(\lambda - 8) = 0
   \]
   Expanding this:
   \[
   \lambda^2 - 12\lambda + 32 = 0
   \]

4. **Compare Coefficients**:
   By comparing the coefficients of the characteristic polynomial:
   - \( \lambda^2 \) term: Coefficient is 1 in both cases.
   - \( \lambda \) term: \( -(2 + y) = -12 \) gives \( y = 10 \).
   - Constant term: \( 2y - 3x = 32 \). Substituting \( y = 10 \):
     \[
     2(10) - 3x = 32 \implies 20 - 3x = 32 \implies 3x = -12 \implies x = -4
     \]

### Conclusion:
The values of \( x \) and \( y \) are \( x = -4 \) and \( y = 10 \).

Would you like more details or have any further questions?

Here are some related questions to expand on the topic:
1. How do you determine the eigenvectors corresponding to the given eigenvalues?
2. What happens to the eigenvalues if the matrix \( A \) is scaled by a constant?
3. Can the trace of the matrix \( A \) be used to find the eigenvalues?
4. What is the relationship between the determinant of a matrix and its eigenvalues?
5. How would the process differ if the matrix \( A \) was not a 2x2 matrix but a 3x3 matrix?

**Tip:** Remember that the sum of the eigenvalues of a matrix is equal to its trace, and the product of the eigenvalues is equal to its determinant.

Explore how to determine the values of x and y in matrix A=[23xy] when its eigenvalues are given as 4 and 8. Understand the characteristic polynomial and eigenvalue equation involved in solving this problem. Suitable for advanced high school students and beyond.

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