If for A = begin{bmatrix} alpha & beta gamma | vedclass

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

Question

(C) Given the matrix $A = \begin{bmatrix} \alpha & \beta \ \gamma & -\alpha \end{bmatrix}$.We are given that $A^2 = I$,where $I$ is the identity matr

Vedclass · Accepted Answer

$1 - \alpha^2 - \beta \gamma = 0$

Vedclass · Answer

(C) Given the matrix $A = \begin{bmatrix} \alpha & \beta \ \gamma & -\alpha \end{bmatrix}$.We are given that $A^2 = I$,where $I$ is the identity matrix $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$.First,calculate $A^2$:$A^2 = \begin{bmatrix} \alpha & \beta \ \gamma & -\alpha \end{bmatrix} \begin{bmatrix} \alpha & \beta \ \gamma & -\alpha \end{bmatrix}$$A^2 = \begin{bmatrix} \alpha^2 + \beta \gamma & \alpha \beta - \beta \alpha \ \gamma \alpha - \alpha \gamma & \beta \gamma + \alpha^2 \end{bmatrix}$$A^2 = \begin{bmatrix} \alpha^2 + \beta \gamma & 0 \ 0 & \alpha^2 + \beta \gamma \end{bmatrix}$Since $A^2 = I$,we have:$\begin{bmatrix} \alpha^2 + \beta \gamma & 0 \ 0 & \alpha^2 + \beta \gamma \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$Comparing the elements,we get $\alpha^2 + \beta \gamma = 1$.Rearranging the terms,we get $1 - \alpha^2 - \beta \gamma = 0$.Therefore,the correct option is $C$.

If for $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$,$A^2 = I$,then . . . . . . .

Similar Questions

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

Compute the indicated products $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$.

If $A = \begin{bmatrix} 1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} -3 & -2 & 4 \\ 2 & 2 & -1 \\ -2 & 0 & 3 \end{bmatrix}$, then $A^2 = $

If $A = \begin{bmatrix} 1 & -2 \\ 4 & 5 \end{bmatrix}$ and $f(t) = t^2 - 3t + 7$, then $f(A) + \begin{bmatrix} 3 & 6 \\ -12 & -9 \end{bmatrix}$ is equal to

Assuming that the sums and products given below are defined,which of the following is not true for matrices?

Vedclass Test Series

Exam Paper Generator

Online Exam Module