If A=left[begin{array}{ll}1 & 1 0 & 1end{arr | vedclass

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

Question

(A) Given $A=\left[\begin{array}{ll}1 & 1 \ 0 & 1\end{array}ight]$ and $I=\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}ight]$.We calculate the

Vedclass · Accepted Answer

$A^n=n A-(n-1) I$

Vedclass · Answer

(A) Given $A=\left[\begin{array}{ll}1 & 1 \ 0 & 1\end{array}ight]$ and $I=\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}ight]$.We calculate the powers of $A$:$A^2 = A \cdot A = \left[\begin{array}{ll}1 & 1 \ 0 & 1\end{array}ight] \left[\begin{array}{ll}1 & 1 \ 0 & 1\end{array}ight] = \left[\begin{array}{ll}1 & 2 \ 0 & 1\end{array}ight] = 2A - I = 2A - (2-1)I$.$A^3 = A^2 \cdot A = \left[\begin{array}{ll}1 & 2 \ 0 & 1\end{array}ight] \left[\begin{array}{ll}1 & 1 \ 0 & 1\end{array}ight] = \left[\begin{array}{ll}1 & 3 \ 0 & 1\end{array}ight] = 3A - 2I = 3A - (3-1)I$.By mathematical induction, we can generalize this pattern for any $n \in N$:$A^n = nA - (n-1)I$.

If $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then for all $n \in N$, find $A^n$.

Similar Questions

If $A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}$ and $B = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$,where $i = \sqrt{-1}$,then which of the following relations is correct?

If $P = \begin{bmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{bmatrix}$ and $X = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}$, then $P^3 X$ is equal to:

Given $A = \begin{bmatrix} \sqrt{3} & 1 & -1 \\ 2 & 3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & \sqrt{5} & 1 \\ -2 & 3 & \frac{1}{2} \end{bmatrix}$,find $A + B = \dots \dots \dots$

If $A$ and $B$ are square matrices of size $n \times n$ such that $A^2 - B^2 = (A - B)(A + B)$,then which of the following will be always true?

If $A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}$,then find the matrix $X$ such that $2A + 3X = 5B$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module