If N∈N and A=a000b000c then An=

A
$(\begin{array}{ccc} 0 & 0 & a^{n} \\ 0 & b^{n} & 0 \\ c^{n} & 0 & 0 \end{array})$
B
$(\begin{array}{ccc} a^{n} & 0 & 0 \\ 0 & b^{n} & 0 \\ 0 & 0 & c^{n} \end{array})$
C
$O$
D
$I$

Solution:

$A = (\begin{array}{lll} a 0 0 \\ 0 b 0 \\ 0 0 c \end{array}) A^{2} = (\begin{array}{lll} a 0 0 \\ 0 b 0 \\ 0 0 c \end{array}) (\begin{array}{lll} a 0 0 \\ 0 b 0 \\ 0 0 c \end{array}) = (\begin{array}{lll} a^{2} 0 0 \\ 0 b^{2} 0 \\ 0 0 c^{2} \end{array})$
$A^{4} = A^{2} A^{2} = (\begin{array}{lll} a^{2} 0 0 \\ 0 b^{2} 0 \\ 0 0 c^{2} \end{array}) (\begin{array}{lll} a^{2} 0 0 \\ 0 b^{2} 0 \\ 0 0 c^{2} \end{array}) = (\begin{array}{lll} a^{4} 0 0 \\ 0 b^{4} 0 \\ 0 0 c^{4} \end{array}) A^{n} = (\begin{array}{lll} a^{n} 0 0 \\ 0 b^{n} 0 \\ 0 0 c^{n} \end{array})$

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

Solution:

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