5.Let A=[(2,3),(x,y)]. If the eigen values are 4 and 8 then find the value of x and y.

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

### Step 1: Write the characteristic equation
The characteristic equation for a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[
\lambda^2 - (\text{Trace of } A)\lambda + \text{Determinant of } A = 0
\]

Where:
- The trace of \( A \) is the sum of the diagonal elements: \( \text{Trace of } A = a + d = 2 + y \).
- The determinant of \( A \) is calculated as: \( \text{Determinant of } A = ad - bc = 2y - 3x \).

Since the eigenvalues are 4 and 8, the characteristic equation must be:

\[
\lambda^2 - (4 + 8)\lambda + (4 \times 8) = 0
\]

Simplifying this:

\[
\lambda^2 - 12\lambda + 32 = 0
\]

### Step 2: Match coefficients with the general form
The general form of the characteristic equation for matrix \( A \) is:

\[
\lambda^2 - (\text{Trace of } A)\lambda + \text{Determinant of } A = 0
\]

Comparing this with the equation \( \lambda^2 - 12\lambda + 32 = 0 \), we get:

\[
\text{Trace of } A = 12 \quad \text{and} \quad \text{Determinant of } A = 32
\]

### Step 3: Set up equations using the trace and determinant
From the trace:

\[
2 + y = 12 \quad \Rightarrow \quad y = 10
\]

From the determinant:

\[
2y - 3x = 32
\]

Substituting \( y = 10 \) into the determinant equation:

\[
2(10) - 3x = 32 \quad \Rightarrow \quad 20 - 3x = 32
\]

Solving for \( x \):

\[
-3x = 32 - 20 \quad \Rightarrow \quad -3x = 12 \quad \Rightarrow \quad x = -4
\]

### Step 4: Final values
The values of \( x \) and \( y \) are:

\[
x = -4, \quad y = 10
\]

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do eigenvalues change if the entries of the matrix are altered?
2. Can you find the eigenvectors corresponding to the given eigenvalues?
3. How does the determinant relate to the eigenvalues of a matrix?
4. What happens to the eigenvalues if you multiply the matrix by a scalar?
5. Can eigenvalues provide insight into the stability of a system represented by a matrix?

**Tip:** When solving for matrix elements using eigenvalues, always cross-check by plugging the results back into the determinant and trace equations to ensure consistency.

Learn how to determine matrix elements x and y from given eigenvalues using the characteristic equation. Detailed step-by-step solution provided with trace and determinant calculations. Suitable for advanced mathematics students.

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