Let A be a 2 times 2 matrix with real entries | vedclass

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

Question

(C) Given $A^2 = I$. Taking the determinant on both sides,we get $\operatorname{det}(A^2) = \operatorname{det}(I) = 1$.Since $\operatorname{det}(A^2)

Vedclass · Accepted Answer

Statement $I$ is true,statement $II$ is false.

Vedclass · Answer

(C) Given $A^2 = I$. Taking the determinant on both sides,we get $\operatorname{det}(A^2) = \operatorname{det}(I) = 1$.Since $\operatorname{det}(A^2) = (\operatorname{det}(A))^2$,we have $(\operatorname{det}(A))^2 = 1$,which implies $\operatorname{det}(A) = 1$ or $\operatorname{det}(A) = -1$.If $\operatorname{det}(A) = 1$,then $A$ is invertible and $A^2 = I$ implies $A = A^{-1}$. For a $2 	imes 2$ matrix,if $\operatorname{det}(A) = 1$ and $A^2 = I$,then $A$ must be $I$ or $-I$.Thus,if $A 
eq I$ and $A 
eq -I$,we must have $\operatorname{det}(A) = -1$. So,Statement $I$ is true.Now consider $A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$. Here $A^2 = I$,$A 
eq I$,and $A 
eq -I$.The trace $\operatorname{Tr}(A) = 0 + 0 = 0$.Since we found a case where $\operatorname{Tr}(A) = 0$ while $A 
eq \pm I$,Statement $II$ is false.

Similar Questions

Let $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$. If $B = I - {}^{5}C_{1} (\operatorname{adj} A) + {}^{5}C_{2} (\operatorname{adj} A)^{2} - \dots - {}^{5}C_{5} (\operatorname{adj} A)^{5}$,then the sum of all elements of the matrix $B$ is

If $A = \begin{bmatrix} \cos \theta & i \sin \theta \\ i \sin \theta & \cos \theta \end{bmatrix}$,$\theta = \frac{\pi}{24}$ and $A^{5} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$,where $i = \sqrt{-1}$,then which one of the following is not true?

Match the items of List-$I$ with those of List-$II$. The correct match is:

In a $\Delta ABC,$ if $\left| \begin{array}{ccc} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{array} \right| = 0$,then $\sin^2 A + \sin^2 B + \sin^2 C = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module