Let A = begin{bmatrix} 1 & frac{1}{51} 0 & 1 | vedclass

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

Question

(A) Let $C = \begin{bmatrix} 1 & 2 \ -1 & -1 \end{bmatrix}$ and $D = \begin{bmatrix} -1 & -2 \ 1 & 1 \end{bmatrix}$.Note that $CD = \begin{bmatrix}

Vedclass · Accepted Answer

$100$

Vedclass · Answer

(A) Let $C = \begin{bmatrix} 1 & 2 \ -1 & -1 \end{bmatrix}$ and $D = \begin{bmatrix} -1 & -2 \ 1 & 1 \end{bmatrix}$.Note that $CD = \begin{bmatrix} 1 & 2 \ -1 & -1 \end{bmatrix} \begin{bmatrix} -1 & -2 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = I$.Since $B = CAD$,we have $B^n = (CAD)(CAD)...(CAD) = CA^n D$.Given $A = \begin{bmatrix} 1 & \frac{1}{51} \ 0 & 1 \end{bmatrix}$,by induction $A^n = \begin{bmatrix} 1 & \frac{n}{51} \ 0 & 1 \end{bmatrix}$.Thus,$B^n = C A^n D = \begin{bmatrix} 1 & 2 \ -1 & -1 \end{bmatrix} \begin{bmatrix} 1 & \frac{n}{51} \ 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & -2 \ 1 & 1 \end{bmatrix}$.$B^n = \begin{bmatrix} 1 & \frac{n}{51} + 2 \ -1 & -\frac{n}{51} - 1 \end{bmatrix} \begin{bmatrix} -1 & -2 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} \frac{n}{51} + 1 & \frac{n}{51} \ -\frac{n}{51} & 1 - \frac{n}{51} \end{bmatrix}$.Summing from $n=1$ to $50$:$\sum_{n=1}^{50} B^n = \begin{bmatrix} \sum_{n=1}^{50} (\frac{n}{51} + 1) & \sum_{n=1}^{50} \frac{n}{51} \ \sum_{n=1}^{50} (-\frac{n}{51}) & \sum_{n=1}^{50} (1 - \frac{n}{51}) \end{bmatrix}$.Using $\sum_{n=1}^{50} n = \frac{50 	imes 51}{2} = 1275$,we get $\sum_{n=1}^{50} \frac{n}{51} = \frac{1275}{51} = 25$.$\sum_{n=1}^{50} B^n = \begin{bmatrix} 25 + 50 & 25 \ -25 & 50 - 25 \end{bmatrix} = \begin{bmatrix} 75 & 25 \ -25 & 25 \end{bmatrix}$.The sum of all elements is $75 + 25 - 25 + 25 = 100$.

List-$I$	List-$II$
$(A)$ If $A$ is a non-singular matrix of order $3$ and $\|A\|=a$,then $\|\text{adj}(A)\|=$	$(I)$ null matrix
$(B)$ $A$ is a non-singular matrix of order $3$ and $B$ is any matrix of order $3$ such that $AB=O$,then $B$ is	$(II)$ $a^2$
$(C)$ $\begin{vmatrix} 1 & x & x^2 \\ \cos(a-b)y & \cos ay & \cos(a+b)y \\ \sin(a-b)y & \sin ay & \sin(a+b)y \end{vmatrix}$ does not depend on	$(III)$ $b$
$(D)$ $A$ is a square matrix of order $3$ and $B=A-A^T$,then $B$ is	$(IV)$ $a$
	$(V)$ $0$

Let $A = \begin{bmatrix} 1 & \frac{1}{51} \\ 0 & 1 \end{bmatrix}$. If $B = \begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$,then the sum of all the elements of the matrix $\sum_{n=1}^{50} B^n$ is equal to

Similar Questions

Let $A^{-1} = \begin{bmatrix} 1 & 2017 & 2 \\ 1 & 2017 & 4 \\ 1 & 2018 & 8 \end{bmatrix}$. Then,$|2A| - |2A^{-1}|$ is equal to

Let $A$ and $B$ be real matrices of the form $\begin{bmatrix} \alpha & 0 \\ 0 & \beta \end{bmatrix}$ and $\begin{bmatrix} 0 & \gamma \\ \delta & 0 \end{bmatrix}$,respectively.
Statement $1$: $AB - BA$ is always an invertible matrix.
Statement $2$: $AB - BA$ is never an identity matrix.

Let $X = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ and $A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}$. For $k \in N$,if $X^{T} A^{k} X = 33$,then $k$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module