Question

If matrix \[A = \left[ {\begin{array}{*{20}{c}}
  0&{ - 1} \\
  1&0
\end{array}} \right]\], then \[{A^{16}} = \]
A. \[\left[ {\begin{array}{*{20}{c}}
  0&{ - 1} \\
  1&0
\end{array}} \right]\]
B. \[\left[ {\begin{array}{*{20}{c}}
  0&1 \\
  1&0
\end{array}} \right]\]
C. \[\left[ {\begin{array}{*{20}{c}}
  { - 1}&0 \\
  0&1
\end{array}} \right]\]
D. \[\left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]\]

Answer 1

Hint: To solve this question we will first find the value of \[{A^2}\]. After this we will calculate \[{A^{4}}\] by multiplying \[{A^2}\] with \[{A^2}\]. After calculating the value of \[{A^{4}}\] we will find the value of find \[{A^16}\] by squaring \[{A^{4}}\] that is by multiplying \[{A^{4}}\] with \[{A^{4}}\].

Formula used:
If \[A = {[{a_{ij}}]_{m \times n}}\] and \[B = {[{b_{ij}}]_{n \times p}}\] then we can say that \[A \times B = C\] where the value of C is
\[C = {[{c_{ij}}]_{m \times p}}\]
Here \[{c_{ij}} = \mathop \sum \limits_{j = 1}^m {a_{ij}}{b_{jk}} = {a_{i1}}{b_{1k}} + {a_{i2}}{b_{2k}} + ........ + {a_{im}}{b_{mk}}\]

Complete step by step Solution:
We are given that,
\[A = \left[ {\begin{array}{*{20}{c}}
  0&{ - 1} \\
  1&0
\end{array}} \right]\]
We will first calculate the value of \[{A^2}\] by multiplying \[A\] with \[A\]
We will now evaluate \[{{A}^{4}}\] by calculating \[{{A}^{4}}={{A}^{2}}\times {{A}^{2}}\].
\[\begin{align}
  & {{A}^{4}}={{A}^{2}}\times {{A}^{2}} \\
 & {{A}^{4}}=\left[ \begin{matrix}
   \text{ }\!\!~\!\!\text{ }-1 & 0 \\
   \text{ }\!\!~\!\!\text{ }0 & -1\text{ }\!\!~\!\!\text{ } \\
\end{matrix} \right]\times \left[ \begin{matrix}
   \text{ }\!\!~\!\!\text{ }-1 & 0 \\
   \text{ }\!\!~\!\!\text{ }0 & -1\text{ }\!\!~\!\!\text{ } \\
\end{matrix} \right] \\
 & {{A}^{4}}=\left[ \begin{matrix}
   \text{ }\!\!~\!\!\text{ }1 & 0 \\
   \text{ }\!\!~\!\!\text{ }0 & 1\text{ }\!\!~\!\!\text{ } \\
\end{matrix} \right]
\end{align}\]
Now we will evaluate \[{{A}^{16}}\] by calculating \[{{A}^{16}}={{A}^{4}}\times {{A}^{4}}\].
\[\begin{align}
  & {{A}^{16}}={{A}^{4}}\times {{A}^{4}} \\
 & {{A}^{16}}=\left[ \begin{matrix}
   \text{ }\!\!~\!\!\text{ }1 & 0 \\
   \text{ }\!\!~\!\!\text{ }0 & 1\text{ }\!\!~\!\!\text{ } \\
\end{matrix} \right]\times \left[ \begin{matrix}
   \text{ }\!\!~\!\!\text{ }1 & 0 \\
   \text{ }\!\!~\!\!\text{ }0 & 1\text{ }\!\!~\!\!\text{ } \\
\end{matrix} \right] \\
 & {{A}^{16}}=\left[ \begin{matrix}
   \text{ }\!\!~\!\!\text{ }1 & 0 \\
   \text{ }\!\!~\!\!\text{ }0 & 1\text{ }\!\!~\!\!\text{ } \\
\end{matrix} \right]
\end{align}\]

Option D. is the correct answer.

Note:To solve the given problem, one must know to multiply two matrices. One must make sure that the terms are added before giving the resultant value in each position of the resultant matrix. One must also know to write large numbers in terms of smaller ones for easier simplification.