If M = begin{bmatrix} frac{5}{2} & frac{3}{2} | vedclass

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

Question

(A) Given $M = \begin{bmatrix} \frac{5}{2} & \frac{3}{2} \ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix}$.We can write $M$ as $M = I + \frac{3}{2} A$,whe

Vedclass · Accepted Answer

$\begin{bmatrix} 3034 & 3033 \ -3033 & -3032 \end{bmatrix}$

Vedclass · Answer

(A) Given $M = \begin{bmatrix} \frac{5}{2} & \frac{3}{2} \ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix}$.We can write $M$ as $M = I + \frac{3}{2} A$,where $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ and $A = \begin{bmatrix} 1 & 1 \ -1 & -1 \end{bmatrix}$.First,calculate $A^2$:$A^2 = \begin{bmatrix} 1 & 1 \ -1 & -1 \end{bmatrix} \begin{bmatrix} 1 & 1 \ -1 & -1 \end{bmatrix} = \begin{bmatrix} 1-1 & 1-1 \ -1+1 & -1+1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O$.Since $A^2 = O$,we can use the binomial expansion for $(I + \frac{3}{2} A)^n$:$M^n = (I + \frac{3}{2} A)^n = I^n + n(I^{n-1})(\frac{3}{2} A) + \frac{n(n-1)}{2} I^{n-2} (\frac{3}{2} A)^2 + \dots$Since $A^2 = O$,all terms involving $A^k$ for $k \ge 2$ are zero.Thus,$M^n = I + n \cdot \frac{3}{2} A$.For $n = 2022$:$M^{2022} = I + 2022 \cdot \frac{3}{2} A = I + 3033 A$.$M^{2022} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} + 3033 \begin{bmatrix} 1 & 1 \ -1 & -1 \end{bmatrix} = \begin{bmatrix} 1+3033 & 0+3033 \ 0-3033 & 1-3033 \end{bmatrix} = \begin{bmatrix} 3034 & 3033 \ -3033 & -3032 \end{bmatrix}$.

If $M = \begin{bmatrix} \frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix}$,then which of the following matrices is equal to $M^{2022}$?

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 $A = \begin{bmatrix} 0 & -\tan(\frac{\theta}{2}) \\ \tan(\frac{\theta}{2}) & 0 \end{bmatrix}$ and $(I_{2} + A)(I_{2} - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$,then $13(a^{2} + b^{2})$ is equal to ...........

If $A+2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}$ and $2A-B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}$, then $\operatorname{tr}(A)-\operatorname{tr}(B) =$

If $A$ and $B$ are two matrices such that $A+B$ and $AB$ are both defined,then

Let $S = \left\{ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} : a_{ij} \in \{0, 1, 2\}, a_{11} = a_{22} \right\}$. Then the number of non-singular matrices in the set $S$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module