If A = begin{bmatrix} 1 & 0 1 & 1 end{bmatri | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $A = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}$. We check the validity of the statement $A^n = \begin{bmatrix} 1 & 0 \ n & 1 \end{bmatri

Vedclass · Accepted Answer

True for $n = 3$

Vedclass · Answer

(C) Given $A = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}$. We check the validity of the statement $A^n = \begin{bmatrix} 1 & 0 \ n & 1 \end{bmatrix}$ for various values of $n \in N$.For $n = 1$: $A^1 = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}$,which matches the formula $\begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}$.For $n = 2$: $A^2 = A \cdot A = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1(1)+0(1) & 1(0)+0(1) \ 1(1)+1(1) & 1(0)+1(1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix}$. This matches the formula for $n = 2$.For $n = 3$: $A^3 = A^2 \cdot A = \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1(1)+0(1) & 1(0)+0(1) \ 2(1)+1(1) & 2(0)+1(1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 3 & 1 \end{bmatrix}$. This matches the formula for $n = 3$.Therefore,the statement is true for $n = 3$.

If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$,then $A^n = \begin{bmatrix} 1 & 0 \\ n & 1 \end{bmatrix}$ for all $n \in N$. Which of the following is correct?

Similar Questions

If $A = [1, 2, 3]$,$B = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}$ and $C = \begin{bmatrix} 1 & 5 \\ 0 & 2 \end{bmatrix}$,then which of the following is defined?

If $2X - \begin{bmatrix} 1 & 2 \\ 7 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 0 & -2 \end{bmatrix}$,then $X$ is equal to

If $A = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix}$ and $f(x) = x + x^2 + x^3 + \ldots + x^{2023}$,then $f(A) + I = $

Evaluate $A^2+2I$ if $A=\begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix}$. (in $A$)

If $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$,then the incorrect option among the following is

Vedclass Test Series

Exam Paper Generator

Online Exam Module