જો શ્રેણિક A = begin{bmatrix} 1 & 0 & 0 0 & | vedclass

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(C) આપેલ છે કે $A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 3 & 0 & -1 \end{bmatrix}$.$A$ ની ઘાતની ગણતરી કરતા:$A^2 = \begin{bmatrix} 1 & 0 & 0 \ 0

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(C) આપેલ છે કે $A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 3 & 0 & -1 \end{bmatrix}$.$A$ ની ઘાતની ગણતરી કરતા:$A^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{bmatrix}$,$A^3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 8 & 0 \ 3 & 0 & -1 \end{bmatrix}$,$A^4 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 16 & 0 \ 0 & 0 & 1 \end{bmatrix}$.ગાણિતિક અનુમાન મુજબ,બેકી $n$ માટે $A^n = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2^n & 0 \ 0 & 0 & 1 \end{bmatrix}$ અને એકી $n$ માટે $A^n = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2^n & 0 \ 3 & 0 & -1 \end{bmatrix}$.તેથી,$A^{20} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2^{20} & 0 \ 0 & 0 & 1 \end{bmatrix}$ અને $A^{19} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2^{19} & 0 \ 3 & 0 & -1 \end{bmatrix}$.સમીકરણ $A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{bmatrix}$ માં કિંમતો મૂકતા:$\begin{bmatrix} 1+\alpha+\beta & 0 & 0 \ 0 & 2^{20}+\alpha 2^{19}+2\beta & 0 \ 3\alpha+3\beta & 0 & 1-\alpha-\beta \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{bmatrix}$.સરખામણી કરતા: $1+\alpha+\beta = 1 \Rightarrow \alpha+\beta = 0 \Rightarrow \beta = -\alpha$.વધુમાં,$2^{20} + \alpha 2^{19} + 2\beta = 4$. $\beta = -\alpha$ મૂકતા:$2^{20} + \alpha 2^{19} - 2\alpha = 4 \Rightarrow \alpha(2^{19}-2) = 4 - 2^{20}$.$\alpha = \frac{4 - 2^{20}}{2^{19}-2} = -2$.તેથી $\beta = 2$.આમ,$\beta - \alpha = 2 - (-2) = 4$.

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