"Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)$$ and B = 7A20 $-$ 20A7 + 2I, where I is an identity matrix of order 3 $\times$ 3. If B = [bij], then b13is equal to _____________."

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Accepted Answer

"Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right) = I + C$$

where, $$I = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right),C = \left( {\matrix{ 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr 0 & 0 & 0 \cr } } \right)$$

$${C^2} = \left( {\matrix{ 0 & 0 & 1 \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right),$$

$${C^3} = \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right) = {C^4} = {C^5} = ........$$

$B = 7{A^{20}} - 20{A^7} + 2I$

$= 7{(I + C)^{20}} + 20{(I + C)^7} + 2I$

$= 7(I + 20C + {}^{20}{C_2}{C^2}) - 20(I + 7C + {}^7{C_2}{C^2}) + 2I$

So

${b_{13}} = 7 \times {}^{20}{C_2}{C^2} - 20 \times {}^7{C_2} = 910$"

Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)$$ and B = 7A²⁰ $-$ 20A⁷ + 2I, where I is an identity matrix of order 3 $\times$ 3. If B = [b_ij], then b₁₃is equal to _____________.

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Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)$$ and B = 7A20 $-$ 20A7 + 2I, where I is an identity matrix of order 3 $\times$ 3. If B = [bij], then b13is equal to _____________.

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Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)$$ and B = 7A²⁰ $-$ 20A⁷ + 2I, where I is an identity matrix of order 3 $\times$ 3. If B = [b_ij], then b₁₃is equal to _____________.