G = left{ begin{bmatrix} x & x x & x end{bma | vedclass

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

Question

(B) Let $A = \begin{bmatrix} x & x \ x & x \end{bmatrix}$. The identity element $E = \begin{bmatrix} e & e \ e & e \end{bmatrix}$ must satisfy $AE =

Vedclass · Accepted Answer

$\begin{bmatrix} 3/4 & 3/4 \ 3/4 & 3/4 \end{bmatrix}$

Vedclass · Answer

(B) Let $A = \begin{bmatrix} x & x \ x & x \end{bmatrix}$. The identity element $E = \begin{bmatrix} e & e \ e & e \end{bmatrix}$ must satisfy $AE = A$.$\begin{bmatrix} x & x \ x & x \end{bmatrix} \begin{bmatrix} e & e \ e & e \end{bmatrix} = \begin{bmatrix} 2xe & 2xe \ 2xe & 2xe \end{bmatrix} = \begin{bmatrix} x & x \ x & x \end{bmatrix}$.Thus,$2xe = x \implies e = 1/2$. The identity element is $E = \begin{bmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$.Let the inverse of $A = \begin{bmatrix} 1/3 & 1/3 \ 1/3 & 1/3 \end{bmatrix}$ be $A^{-1} = \begin{bmatrix} y & y \ y & y \end{bmatrix}$.Then $AA^{-1} = E \implies \begin{bmatrix} 1/3 & 1/3 \ 1/3 & 1/3 \end{bmatrix} \begin{bmatrix} y & y \ y & y \end{bmatrix} = \begin{bmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$.Multiplying the matrices: $\begin{bmatrix} 2y/3 & 2y/3 \ 2y/3 & 2y/3 \end{bmatrix} = \begin{bmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$.Equating the elements: $2y/3 = 1/2 \implies y = 3/4$.Therefore,$A^{-1} = \begin{bmatrix} 3/4 & 3/4 \ 3/4 & 3/4 \end{bmatrix}$.

$G = \left\{ \begin{bmatrix} x & x \\ x & x \end{bmatrix} : x \in \mathbb{R} \setminus \{0\} \right\}$ is a group with respect to matrix multiplication. In this group,the inverse of $\begin{bmatrix} 1/3 & 1/3 \\ 1/3 & 1/3 \end{bmatrix}$ is

Similar Questions

If $A=\begin{bmatrix} b & a & 0 \\ c & 0 & b \\ a & a & b \end{bmatrix}$ and $B=\begin{bmatrix} 0 & a & b \\ b & 0 & c \\ b & a & a \end{bmatrix}$ are two matrices such that $AB=\begin{bmatrix} 2 & 2 & 7 \\ 1 & 8 & 5 \\ 3 & 6 & 10 \end{bmatrix}$, then $a^2+b^2+c^2=$

If $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$,then the incorrect option among the following is

If $A$ is a $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime}B$ and $BA^{\prime}$ are both defined,then $B$ is of the type:

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

If $A = [1, 2, 3]$,$B = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}$ and $C = \begin{bmatrix} 1 & 5 \\ 0 & 2 \end{bmatrix}$,then which of the following is defined?

Vedclass Test Series

Exam Paper Generator

Online Exam Module