Question

A real \( 2 \times 2 \) non-singular matrix \( A \) with repeated eigenvalue is given as 

where \( x \) is a real positive number. The value of \( x \) (rounded off to one decimal place) is _________. 
 

Hint:

For a matrix to have repeated eigenvalues, its determinant and trace must satisfy specific conditions that can be solved using the characteristic equation.

Answer 1

Correct Answer: 10

For a matrix to have repeated eigenvalues, its determinant and trace must be the same. The eigenvalue \( \lambda \) of the matrix is given by the characteristic equation: \[ \text{det}(A - \lambda I) = 0 \] The characteristic equation for the matrix is: 

This simplifies to: \[ (x - \lambda)(4 - \lambda) + 9 = 0 \] Solving for \( x \) using the condition that the eigenvalue is repeated (i.e., the discriminant is zero), we find: \[ x = 10.0 \] Thus, the value of \( x \) is \( 10.0 \).