The characteristic equation of a 3 × 3 matrix P is defined as: |
The characteristic equation of a 3 × 3 matrix P is defined as:
|λI - P| = λ3 + λ2 + 2λ + 1 = 0
“I” denotes identity matrix, then inverse of matrix P will be:A. P<sup>2</sup> + P + 2I
B. P<sup>2</sup> + P + I
C. -(P<sup>2</sup> + P + I)
D. -(P<sup>2</sup> + P + 2I)
Please scroll down to see the correct answer and solution guide.
Right Answer is: D
SOLUTION
Concept:
Cayley-Hamilton theorem: According to the Cayley-Hamilton theorem, every matrix 'A' satisfies its own characteristic equation.
Characteristic equation: 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.
Calculation:
|λI - P| = λ3 + λ2 + 2λ + 1 = 0
By using Cayley-Hamilton theorem,
P3 + P2 + 2P + 1 = 0
By multiplying with P-1 on both sides,
P2 + P + 2I + P-1 = 0
⇒ P-1 = – (P2 + P + 2I)