Q.33 The positive Eigen value of the following matrix is ____________.
| 2 1 | | 5 -2 |
The positive eigenvalue of the given 2×2 matrix is
3. Below is a complete step-by-step derivation
using the characteristic equation, along with verification and
practical relevance for engineering mathematics exams.
Given Matrix
Let
A =
(
2 1
5 −2
)
Characteristic Equation Derivation
Eigenvalues satisfy the characteristic equation:
det(A − λI) = 0
where I is the 2×2 identity matrix.
A − λI =
(
2 − λ 1
5 −2 − λ
)
Taking the determinant:
(2 − λ)(−2 − λ) − (1)(5)
Expanding:
= (−4 − 2λ + 2λ + λ2) − 5
= λ2 − 9
Hence, the characteristic equation is:
λ2 − 9 = 0
Eigenvalues Calculation
Solving the equation:
λ2 − 9 = 0
we get:
- λ = 3
- λ = −3
Therefore, the positive eigenvalue of the matrix is:
λ = 3
Why Positive Eigenvalue Matters
In engineering applications such as stability analysis, vibrations,
and control systems:
- Positive eigenvalues indicate growing or unstable modes
- Negative eigenvalues represent decay or stable behavior
This matrix has one positive eigenvalue (3) and one negative eigenvalue (−3),
indicating mixed stability characteristics.
Verification Steps
Substituting λ = 3:
A − 3I =
(
−1 1
5 −5
)
The rank of this matrix is less than 2, confirming the existence
of a non-trivial solution and validating λ = 3 as an eigenvalue.
Numerical computation (e.g., using NumPy) also yields eigenvalues
approximately [3, −3].
