If \(A = \begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}\) and I is the identity matrix of order 3, then \(A^{100}\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A
Let us find the first few powers of A by row-by-column multiplication:
\( \displaystyle A^2 = \begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix} \)
\( \displaystyle A^3 = A^2A = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} = I \)
Since \(100 = 3 \times 33 + 1\),
\(A^{100} = (A^3)^{33}A\)
\(= I^{33}A = A\)