If A = ⎡ ⎢ ⎣ a 0 0 0 b 0 0 0 c ⎤ ⎥ ⎦ , then A n = Options: A . &nbsp ⎡ ⎢ ⎣ n a 0 0 0 n b 0 0 0 n c ⎤ ⎥ ⎦ B . &nbsp ⎡ ⎢ ⎣ a 0 0 0 b 0 0 0 c ⎤ ⎥ ⎦ C . &nbsp ⎡ ⎢ ⎣ a n 0 0 0 b n 0 0 0 c n ⎤ ⎥ ⎦ D . &nbsp N o n e o f t h e s e

Answer: Option C : C Since A 2 = A . A = ⎡ ⎢ ⎣ a 0 0 0 b 0 0 0 c ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ a 0 0 0 b 0 0 0 c ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ a 2 0 0 0 b 2 0 0 0 c 2 ⎤ ⎥ ⎦ And A 3 = ⎡ ⎢ ⎣ a 3 0 0 0 b 3 0 0 0 c 3 ⎤ ⎥ ⎦ , . . . . ⇒ A n = A n − 1 . A = ⎡ ⎢ ⎣ a n − 1 0 0 0 b n − 1 0 0 0 c n − 1 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ a 0 0 0 b 0 0 0 c ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ a n 0 0 0 b n 0 0 0 c n ⎤ ⎥ ⎦ . Note: Students should remember this question as a formula.

If A=⎡⎢⎣a000b000c⎤⎥⎦, then An=

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Question -

If $A = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦$ , then $A^{n} =$

Options:

A .

⎡ ⎢ ⎣ \begin{matrix} n a & 0 & 0 0 & n b & 0 0 & 0 & n c \end{matrix} ⎤ ⎥ ⎦

B .

⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦

C .

⎡ ⎢ ⎣ \begin{matrix} a^{n} & 0 & 0 0 & b^{n} & 0 0 & 0 & c^{n} \end{matrix} ⎤ ⎥ ⎦

D .

N o n e o f t h e s e

Answer: Option C
:
C
Since

A^{2} = A . A = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \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} ⎤ ⎥ ⎦

And

A^{3} = ⎡ ⎢ ⎣ \begin{matrix} a^{3} & 0 & 0 0 & b^{3} & 0 0 & 0 & c^{3} \end{matrix} ⎤ ⎥ ⎦, . . . . \Rightarrow A^{n} = A^{n - 1} . A = ⎡ ⎢ ⎣ \begin{matrix} a^{n - 1} & 0 & 0 0 & b^{n - 1} & 0 0 & 0 & c^{n - 1} \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a^{n} & 0 & 0 0 & b^{n} & 0 0 & 0 & c^{n} \end{matrix} ⎤ ⎥ ⎦ .

Note: Students should remember this question as a formula.

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