If M = begin{bmatrix} 1 & 2 2 & 3 end{bmatri | vedclass

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

Question

(D) Given the equation $M^2 - \lambda M - I_2 = 0$,where $M = \begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix}$.First,calculate $M^2$:$M^2 = \begin{bmatri

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(D) Given the equation $M^2 - \lambda M - I_2 = 0$,where $M = \begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix}$.First,calculate $M^2$:$M^2 = \begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix} = \begin{bmatrix} (1)(1) + (2)(2) & (1)(2) + (2)(3) \ (2)(1) + (3)(2) & (2)(2) + (3)(3) \end{bmatrix} = \begin{bmatrix} 5 & 8 \ 8 & 13 \end{bmatrix}$.Substitute $M^2$,$M$,and $I_2$ into the equation:$\begin{bmatrix} 5 & 8 \ 8 & 13 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$.This simplifies to:$\begin{bmatrix} 5 - \lambda - 1 & 8 - 2\lambda - 0 \ 8 - 2\lambda - 0 & 13 - 3\lambda - 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$.Equating the elements:$4 - \lambda = 0 \Rightarrow \lambda = 4$.$8 - 2\lambda = 0 \Rightarrow 2\lambda = 8 \Rightarrow \lambda = 4$.$12 - 3\lambda = 0 \Rightarrow 3\lambda = 12 \Rightarrow \lambda = 4$.Since $\lambda = 4$ satisfies all equations,the correct option is $D$.

If $M = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ and $M^2 - \lambda M - I_2 = 0$,then $\lambda = $

Similar Questions

If the matrices $A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & 1 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 5 & 0 \end{bmatrix}$, then $AB$ will be

If $A = [1\, 2\, 3]$ and $B = \begin{bmatrix} -5 & 4 & 0 \\ 0 & 2 & -1 \\ 1 & -3 & 2 \end{bmatrix}$,then $AB = $

$AB = 0$,if and only if

Find the values of $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfies the equation $A^{\prime} A = I$.

If $A = \begin{bmatrix} 4 & x + 2 \\ 2x - 3 & x + 1 \end{bmatrix}$ is a symmetric matrix,then $x = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module