eigen value $\lambda^{*}$ of matrix A is said to have arithmetic multiplicity 'm'.(solving characteristic equation m times eigen value repeating. )
Theorem: if $\lambda$ is an eigen value of A then multiplicity of $\lambda^{*}$ is at least the dimension of the eigenspace $E_{\lambda^*}$.$[E_{\lambda}=${X| AX=$\lambda$X } is called eigen space of A associated with $\lambda$]
The matrix A is a 3 × 3 matrix, so it has 3 eigenvalues in total. The eigenspace $E_7$ contains the vectors $(1, 2, 1)^T $and $(1, 1, 0)^T$ , which are linearly independent. So$E_7$ must have dimension at least 2, which implies that the eigenvalue 7 has multiplicity at least 2.
Let the other eigenvalue be λ, then from the trace λ+7+7 = 2, so λ = −12. So the three eigenvalues are 7, 7 and -12. Hence, the determinant of A is 7 × 7 × −12 = −588.
