Q18Mathematics II

Question

PART - C

Q.1 Verify Cayley Hamilton theorem for matrix -

A = 102020213

and hence find its inverse.

Answer 1

The Cayley-Hamilton theorem absolutely asserts that every real square matrix rigidly satisfies its very own characteristic equation.

Step 1: Characteristic Equation

We strictly evaluate the mathematical determinant $∣ A - λ I ∣ = 0$ :

1 - λ 02 0 2 - λ 0 21 3 - λ = 0

Expanding rigidly exactly along the first row:

(1 - λ) [(2 - λ) (3 - λ) - 0] - 0 + 2 [0 - 2 (2 - λ)] = 0

(1 - λ) (λ^{2} - 5 λ + 6) - 4 (2 - λ) = 0

(λ^{2} - 5 λ + 6 - λ^{3} + 5 λ^{2} - 6 λ) - 8 + 4 λ = 0

- λ^{3} + 6 λ^{2} - 7 λ - 2 = 0 ⟹ λ^{3} - 6 λ^{2} + 7 λ + 2 = 0

Step 2: Verifying Cayley-Hamilton

We must strictly prove mathematically that $A^{3} - 6 A^{2} + 7 A + 2 I = 0$ .

Calculate exactly $A^{2}$ :

A^{2} = 102020213102020213 = 5280408513

Calculate exactly $A^{3}$ :

A^{3} = A^{2} A = 5280408513102020213 = 211234080342355

Substitute these completely into the matrix equation:

A^{3} - 6 A^{2} + 7 A + 2 I = 211234080342355 - 3012480240483078 + 7014014014721 + 200020002

Summing the explicit elements for the $(1, 1)$ position: $21 - 30 + 7 + 2 = 0$ . This rigorously holds absolutely exactly for all 9 elements. Thus, the theorem is mathematically verified.

Step 3: Calculating the Inverse

Multiply the entire proven identity exactly by $A^{- 1}$ :

A^{2} - 6 A + 7 I + 2 A^{- 1} = 0 ⟹ A^{- 1} = - \frac{1}{2} (A^{2} - 6 A + 7 I)

A^{- 1} = - \frac{1}{2} 5280408513 - 6012012012618 + 700070007

A^{- 1} = - \frac{1}{2} 62 - 4 0 - 1 0 - 4 - 1 2 = - 3 - 1 2 0 1/2 0 2 1/2 - 1

Q18Mathematics II

Question

Answer

Step 1: Characteristic Equation

Step 2: Verifying Cayley-Hamilton

Step 3: Calculating the Inverse