Question

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:
1. P² + P + 2I
2. P² + P + I
3. -(P² + P + I)
4. -(P² + P + 2I)

Answer 1

Correct Answer - Option 4 : -(P² + P + 2I)

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)