If A is a matrix {bmatrix} 1 & 0 \\ 1 & 1 {bmatrix}, then Aⁿ = {bmatrix} 1 & 0 \\ n & 1 {bmatrix}, n N

Correct Answer is : true for n=3n=3n=3

Correct Answer is : true for n=3n=3n=3

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AP EAPCET 25-Aug-2021 Shift 2 Paper

Section: Mathematics

Question : 4 of 160

Marks: +1, -0

A

\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}

A^{n} = \begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix}, \forall n \in N

Solution: 👈: Video Solution

Given, If A is a matrix

\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}

, then

A^{n} = \begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix}, \forall n \in N

We will generalise the exponent of A.

Let

n=2

∴

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

= \begin{bmatrix} 1.1+0.1 & 1.0+.01 \\ 1.1+1.1 & 1.0+1.1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}

Again, let

n=3

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

Similarly,

A^{4} = \begin{bmatrix} 1 & 0 \\ 4 & 1 \end{bmatrix}

Hence,

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

is true for

n=3

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