If n∈N and A=(a000b000c) then An=

A=(a000b000c) A2=(a000b000c)×[a000b000c]=[a2000b2000c2] A3=[a2000b2000c2]×[a000b000c]=[a3000b3000b3] ∴An=(an000bn000cn)

If n∈N and A=(a000b000c) then An=

A=(a000b000c) A2=(a000b000c)×[a000b000c]=[a2000b2000c2] A3=[a2000b2000c2]×[a000b000c]=[a3000b3000b3] ∴An=(an000bn000cn)

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If $n \in N$ and $A = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})$ then $A^{n} =$

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(\begin{matrix} 0 & 0 & a^{n} \\ 0 & b^{n} & 0 \\ c^{n} & 0 & 0 \end{matrix})

(\begin{matrix} a^{n} & 0 & 0 \\ 0 & b^{n} & 0 \\ 0 & 0 & c^{n} \end{matrix})

answer is B.

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Detailed Solution

A = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})

$A^{2} = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}) \times [\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}] = [\begin{matrix} a^{2} & 0 & 0 \\ 0 & b^{2} & 0 \\ 0 & 0 & c^{2} \end{matrix}]$

$A^{3} = [\begin{matrix} a^{2} & 0 & 0 \\ 0 & b^{2} & 0 \\ 0 & 0 & c^{2} \end{matrix}] \times [\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}] = [\begin{matrix} a^{3} & 0 & 0 \\ 0 & b^{3} & 0 \\ 0 & 0 & b^{3} \end{matrix}]$