Let $A = \begin{bmatrix} 1 & -1 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1 \end{bmatrix}$ and $B = 7A^{20} - 20A^{7} + 2I$, where $I$ is an identity matrix of order $3 imes 3$. If $B = [b_{ij}]$, then $b_{13}$ is equal to.

Question

Let $A = \begin{bmatrix} 1 & -1 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1 \end{bmatrix}$ and $B = 7A^{20} - 20A^{7} + 2I$, where $I$ is an identity matrix of order $3 	imes 3$. If $B = [b_{ij}]$, then $b_{13}$ is equal to.

Aicharya · Accepted Answer

$A = \begin{bmatrix} 1 & -1 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1 \end{bmatrix} = I + C$ where $I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$, $C = \begin{bmatrix} 0 & -1 & 0 \ 0 & 0 & -1 \ 0 & 0 & 0 \end{bmatrix}$ Then, $C^2 = \begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}$, $C^3 = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} = C^4 = C^5 = \ldots$ Now, $B = 7A^{20} - 20A^7 + 2I$ $= 7(I + C)^{20} - 20(I + C)^7 + 2I$ $= 7[I + 20C + C^{20} + \ldots] - 20[I + 7C + C^{7} + \ldots] + 2I$ $= -11I + 910C^2$ $= \begin{bmatrix} -11 & 0 & 910 \ 0 & -11 & 0 \ 0 & 0 & -11 \end{bmatrix}$ So, $b_{13} = 910$

Let $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$ and $B = 7A^{20} - 20A^{7} +…

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