Question

The solution of $\det(A - \lambda I_2) = 0$ is $4$ and $8$, and $A = \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix}$. Then

• A. x= 4, y= 10
• B. x=5, y=8
• C. x=3, y=9
• D. x= -4, y=10

Answer 1

The Correct Option is D

We are given the matrix \( A = \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix} \) and the eigenvalues of \( A \) are \( 4 \) and \( 8 \). To find the values of \( x \) and \( y \), we use the fact that the eigenvalues are the solutions to the characteristic equation: \[ \det(A - \lambda I_2) = 0 \] where \( \lambda \) is the eigenvalue and \( I_2 \) is the \( 2 \times 2 \) identity matrix.

Step 1: Write the characteristic equation
The characteristic equation is given by: \[ \det\left( \begin{pmatrix} 2 & 3 \\ x & y \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) = 0 \] This simplifies to: \[ \det\begin{pmatrix} 2 - \lambda & 3 \\ x & y - \lambda \end{pmatrix} = 0 \] The determinant of a \( 2 \times 2 \) matrix is calculated as: \[ (2 - \lambda)(y - \lambda) - 3x = 0 \] Expanding this, we get: \[ (2 - \lambda)(y - \lambda) - 3x = 2y - 2\lambda - \lambda y + \lambda^2 - 3x = 0 \] 

Step 2: Use the known eigenvalues
The eigenvalues are \( \lambda = 4 \) and \( \lambda = 8 \). Substituting these values one by one into the equation: - For \( \lambda = 4 \): \[ (2 - 4)(y - 4) - 3x = 0 \quad \Rightarrow \quad -2(y - 4) - 3x = 0 \quad \Rightarrow \quad -2y + 8 - 3x = 0 \] \[ 3x + 2y = 8 \] - For \( \lambda = 8 \): \[ (2 - 8)(y - 8) - 3x = 0 \quad \Rightarrow \quad -6(y - 8) - 3x = 0 \quad \Rightarrow \quad -6y + 48 - 3x = 0 \] \[ 3x + 6y = 48 \] 

Step 3: Solve the system of equations
We now have the system of linear equations: \[ 3x + 2y = 8 \quad \text{(1)} \] \[ 3x + 6y = 48 \quad \text{(2)} \] Subtract equation (1) from equation (2): \[ (3x + 6y) - (3x + 2y) = 48 - 8 \] \[ 4y = 40 \quad \Rightarrow \quad y = 10 \] Substitute \( y = 10 \) into equation (1): \[ 3x + 2(10) = 8 \quad \Rightarrow \quad 3x + 20 = 8 \quad \Rightarrow \quad 3x = -12 \quad \Rightarrow \quad x = -4 \]

Answer:

\[ \boxed{x = -4, y = 10} \]