Correct Answer - Option 4 : -1, -2, -3
Concept:
If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix. The determinant of this matrix equated to zero i.e. |A – λI| = 0 is called the characteristic equation of A.
The roots of the characteristic equation are called Eigenvalues or latent roots or characteristic roots of matrix A.
Properties of Eigenvalues:
The sum of Eigenvalues of a matrix A is equal to the trace of that matrix A
The product of Eigenvalues of a matrix A is equal to the determinant of that matrix A
Calculation:
From the given state-space representation, the system matrix is
\(A = \left[ {\begin{array}{*{20}{c}}
0&1&0\\
3&0&2\\
{ - 12}&{ - 7}&{ - 6}
\end{array}} \right]\)
\(\left[ {sI - A} \right] = \left[ {\begin{array}{*{20}{c}}
s&{ - 1}&0\\
{ - 3}&s&{ - 2}\\
{12}&7&{s + 6}
\end{array}} \right]\)
\(\left| {sI - A} \right| = \left| {\begin{array}{*{20}{c}}
s&{ - 1}&0\\
{ - 3}&s&{ - 2}\\
{12}&7&{s + 6}
\end{array}} \right| = 0\)
⇒ s [ s(s + 6) + 14] + 1 [-3(s + 6) + 24] = 0
⇒ s3 + 6s2 + 14s – 3s – 18 + 24 = 0
⇒ s3 + 6s2 + 11s + 6 = 0
⇒ s = -1, -2, -3
Alternate Method:
Given matrix, \(A = \left[ {\begin{array}{*{20}{c}}
0&1&0\\
3&0&2\\
{ - 12}&{ - 7}&{ - 6}
\end{array}} \right]\)
Sum of Eigen values of a matrix A = Trace of the matrix A = 0 + 0 - 6 = -6
Clearly, only option 2 holds this true.