For a $2 \times 2$ matrix $A$, whose elements are real numbers, denote by $A^m$ the product $AA\dots A$ ($m$ times), where $m$ is a positive integer. Define $x_0 = 0$, $x_1 = 1$, $x_n = x_{n-1} + x_{n-2}$, for all $n \ge 2$ and \[ A_n = \begin{bmatrix} x_{n+1} & x_n\\ x_n & x_{n-1} \end{bmatrix}, \text{ for all } n \ge 1. \] Which of the following statements is TRUE for all $m \ge 3$?

Question

Let the foot of the perpendicular from the point $ (1, 2, 4) $ on the line $ \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} $ be $ P $. Then the distance of $ P $ from the plane $ 3x + 4y + 12z + 23 = 0 $ is:

Answer 1

Step 1: Understanding the Concept:
The sequence $ x_{n} $ is the standard Fibonacci sequence starting with $ x_{0} = 0 $ and $ x_{1} = 1 $.
The matrix $ A_{1} $ is defined as $ \begin{bmatrix} x_{2} & x_{1} x_{1} & x_{0} \end{bmatrix} $.
Key Formula or Approach:
We use the Cayley-Hamilton Theorem, which states that every square matrix satisfies its own characteristic equation.

Step 2: Detailed Explanation:
1. Determine $ A_{1} $:
Given $ x_{0} = 0, x_{1} = 1 $, then $ x_{2} = x_{1} + x_{0} = 1 $.
\[ A_{1} = \begin{bmatrix} 1 & 1 1 & 0 \end{bmatrix} \]
2. Find Characteristic Equation:
The characteristic equation is $ \det(A_{1} - \lambda I) = 0 $:
\[ \det \begin{bmatrix} 1-\lambda & 1 1 & -\lambda \end{bmatrix} = (1-\lambda)(-\lambda) - 1 = 0 \implies \lambda^{2} - \lambda - 1 = 0 \]
3. Apply Cayley-Hamilton Theorem:
Substituting $ A_{1} $ for $ \lambda $:
\[ A_{1}^{2} - A_{1} - I = 0 \implies A_{1}^{2} = A_{1} + I \]
4. Generalize for power $ m $:
Multiply both sides of the identity by $ A_{1}^{m-2} $ (valid for $ m \ge 2 $):
\[ A_{1}^{2} \cdot A_{1}^{m-2} = (A_{1} + I) \cdot A_{1}^{m-2} \implies A_{1}^{m} = A_{1}^{m-1} + A_{1}^{m-2} \]

Step 3: Final Answer:
The recurrence relation $ A_{1}^{m} = A_{1}^{m-1} + A_{1}^{m-2} $ holds true for all $ m \ge 3 $.

Show Hint

The Correct Option is A

Solution and Explanation

Top Questions on Matrices

Questions Asked in IISER exam