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 _________.
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 _________.
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Correct Answer: 10
Solution and Explanation
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 \( 10.0 \).
Top Questions on Eigenvalues
- The eigenvalues of the matrix \[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix} \] are \(\lambda_1, \lambda_2, \lambda_3\). The value of \(\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)\) is:
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The eigenvalues of the matrix
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(Answer in integer)
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