"Let $A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$ where $i = \sqrt { - 1}$. Then, the number of elements in the set { n $\in$ {1, 2, ......, 100} : An = A } is ____________."

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Accepted Answer

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$\therefore$ $${A^2} = \left[ {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right]\left[ {\matrix{ {1 + i} & 1 \cr { - 1} & 0 \cr } } \right] = \left[ {\matrix{ i & {1 + i} \cr {1 - i} & { - i} \cr } } \right]$$

$${A^4} = \left[ {\matrix{ i & {1 + i} \cr {1 - i} & { - i} \cr } } \right]\left[ {\matrix{ i & {1 + i} \cr {1 - i} & { - i} \cr } } \right] = I$$

So A⁵ = A, A⁹ = A and so on.

Clearly n = 1, 5, 9, ......, 97

Number of values of n = 25

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Let $A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$ where $i = \sqrt { - 1}$. Then, the number of elements in the set { n $\in$ {1, 2, ......, 100} : Aⁿ = A } is ____________.

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Let $A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$ where $i = \sqrt { - 1}$. Then, the number of elements in the set { n $\in$ {1, 2, ......, 100} : An = A } is ____________.

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More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.

Let $A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$ where $i = \sqrt { - 1}$. Then, the number of elements in the set { n $\in$ {1, 2, ......, 100} : Aⁿ = A } is ____________.