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A system is described by

\(\dot x\left( t \right) = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 3&0&2\\ { - 12}&{ - 7}&{ - 6} \end{array}} \right]x\left( t \right) + \left[ {\begin{array}{*{20}{c}} 1\\ 0\\ 2 \end{array}} \right]u\left( t \right)\)

\(y\left( t \right) = \left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right]x\left( t \right)\)

Its Eigenvalues are
1. 1, 2, 3
2. 2, 2, 3
3. 12, 6, 7
4. -1, -2, -3

1 Answer

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Best answer
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.

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