Question:
If 3 and 6 are eigenvalues of the matrix \[ \begin{pmatrix} 5 & 2 & 0 \\ 2 & \mu & 0 \\ -3 & 4 & 6 \end{pmatrix} \] then the value of \( \mu \) is \(\underline{\hspace{2cm}}\).
If 3 and 6 are eigenvalues of the matrix \[ \begin{pmatrix} 5 & 2 & 0 \\ 2 & \mu & 0 \\ -3 & 4 & 6 \end{pmatrix} \] then the value of \( \mu \) is \(\underline{\hspace{2cm}}\).
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For matrices, use the characteristic equation to solve for eigenvalues and determine unknown elements like \( \mu \).
Updated On: Jan 7, 2026
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Correct Answer: 5
Solution and Explanation
The determinant of the matrix is calculated by the characteristic equation, which is derived by finding the eigenvalues of the matrix. Given that the eigenvalues of the matrix are 3 and 6, we can substitute them into the characteristic equation and solve for \( \mu \). After solving, we find:
\[
\mu = 5.
\]
Thus, the value of \( \mu \) is \( 5 \).
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