Let the matrix A = begin{bmatrix} 1 & 0 & 0 | vedclass

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

Question

(A) Given $A^n = A^{n-2} + A^2 - I$. For $n=50$,we have $A^{50} = A^{48} + A^2 - I$.By repeating this recurrence relation,we get $A^{50} = A^{48} + (A

Vedclass · Accepted Answer

$53$

Vedclass · Answer

(A) Given $A^n = A^{n-2} + A^2 - I$. For $n=50$,we have $A^{50} = A^{48} + A^2 - I$.By repeating this recurrence relation,we get $A^{50} = A^{48} + (A^2 - I) = A^{46} + 2(A^2 - I) = A^{44} + 3(A^2 - I) = \dots = A^2 + 24(A^2 - I) = 25A^2 - 24I$.First,calculate $A^2$: $A^2 = \begin{bmatrix} 1 & 0 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 0 & 1 \end{bmatrix}$.Now,$A^{50} = 25 \begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 1 & 0 & 1 \end{bmatrix} - 24 \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 25-24 & 0 & 0 \ 25 & 25-24 & 0 \ 25 & 0 & 25-24 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 25 & 1 & 0 \ 25 & 0 & 1 \end{bmatrix}$.The sum of all elements is $1 + 0 + 0 + 25 + 1 + 0 + 25 + 0 + 1 = 53$.

Let the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ satisfy $A^n = A^{n-2} + A^2 - I$ for $n \geq 3$. Then the sum of all the elements of $A^{50}$ is:

Similar Questions

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 $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -5 & 7 & 1 \\ 1 & -5 & 7 \\ 7 & 1 & -5 \end{bmatrix}$,then $AB$ is equal to

For matrices $X$ and $Y$,if $X+Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$ and $X-Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$,then $2X =$ . . . . . .

For a matrix $A$,the conditions $AI = A$ and $AA^T = I$ are true for:

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^{2}$ is $1,$ then the possible number of such matrices is

Vedclass Test Series

Exam Paper Generator

Online Exam Module