Which one of the following is not an eigen value of the matrix \(\begin{bmatrix} -1 & 2 & 2 \\ 2 & 2 & 2 \\ -3 & -6 & -6 \end{bmatrix}\) ?
UGC NET - 2013,
Answer: D
3 is not an eigenvalue of the matrix.
The trace of the matrix is \(-1+2-6=-5\), the determinant is 0, and the sum of the principal 2 by 2 minors is \(-6+12+0=6\). The characteristic equation is therefore
\[ \lambda^{3}+5\lambda^{2}+6\lambda = 0 \]
\[ \lambda(\lambda+2)(\lambda+3)=0 \]
The eigenvalues are 0, -2 and -3, so 0, -2 and -3 are eigenvalues, and 3 is not.
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