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 $f(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & -\cos \theta \end{bmatrix}$,then $f\left(\frac{\pi}{6}\right) = $ . . . . . . .

If $P = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} -1 & -2 \\ -2 & 0 \\ 0 & -4 \end{bmatrix} \begin{bmatrix} -4 & -5 & -6 \\ 0 & 0 & 1 \end{bmatrix}$,then $P_{22} = $

Let $A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Then the number of $3 \times 3$ matrices $B$ with entries from the set $\{1, 2, 3, 4, 5\}$ and satisfying $AB = BA$ is $....$

The identity element in the group $M = \left\{ \begin{bmatrix} x & x \\ x & x \end{bmatrix} \mid x \in \mathbb{R}, x \neq 0 \right\}$ with respect to matrix multiplication is

Let $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$ and $B = 7A^{20} - 20A^{7} + 2I$,where $I$ is an identity matrix of order $3 \times 3$. If $B = [b_{ij}]$,then $b_{13}$ is equal to $....$

Vedclass Test Series

Exam Paper Generator

Online Exam Module