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Question: If A = \(\begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix}\), then A<sup>n</sup> (where n Ī N) is...

If A = [3−41−1]\begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix}, then An (where n Ī N) is

A

[3n−4nn−n]\begin{bmatrix} 3n & - 4n \\ n & - n \end{bmatrix}

B

[n+25−nn−n]\begin{bmatrix} n + 2 & 5 - n \\ n & - n \end{bmatrix}

C

[3n(−4)n1(−1)n]\begin{bmatrix} 3^{n} & ( - 4)^{n} \\ 1 & ( - 1)^{n} \end{bmatrix}

D

None of these

Answer

None of these

Explanation

Solution

We have

A2 = [3−41−1]\begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix} [3−41−1]\begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix}= [5−82−3]\begin{bmatrix} 5 & - 8 \\ 2 & - 3 \end{bmatrix}For n = 2, none of (1) , (2) , (3) match with the actual answer.

Thus, answer is (4).