NIMCET 2012 — Mathematics PYQ

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Question

NIMCET 2012 — Mathematics PYQ

NIMCET | Mathematics | 2012

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ , then $A^n$ for any natural number $n$ is:

Choose the correct answer:

Explanation

1. Calculate $A^2$ :

A^2 = A \cdot A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}

A^2 = \begin{bmatrix} (1 \cdot 1 + 1 \cdot 0) & (1 \cdot 1 + 1 \cdot 1) \\ (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

2. Calculate $A^3$ :

A^3 = A^2 \cdot A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}

A^3 = \begin{bmatrix} (1 \cdot 1 + 2 \cdot 0) & (1 \cdot 1 + 2 \cdot 1) \\ (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}

3. Observe the Pattern:

For $n = 1$ : $A^1 = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$
For $n = 2$ : $A^2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$
For $n = 3$ : $A^3 = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$

By mathematical induction, for any natural number $n$ , the element at $a_{12}$ (first row, second column) becomes $n$ , while the other elements remain unchanged.

General Formula:

A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}

Correct Option:

(b) $\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$