Question

If $A=\left( \begin{matrix}
   0 & 1 \\
   1 & 0 \\
\end{matrix} \right)$ then ${{A}^{4}}$ is equal to
A . $\left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right)$
B. $\left( \begin{matrix}
   1 & 1 \\
   0 & 0 \\
\end{matrix} \right)$
C. $\left( \begin{matrix}
   0 & 0 \\
   1 & 1 \\
\end{matrix} \right)$
D. $\left( \begin{matrix}
   0 & 1 \\
   1 & 0 \\
\end{matrix} \right)$

Answer 1

Hint: In this question, we have given a matric A and we have to find out the value of ${{A}^{4}}$. We know ${{A}^{4}}$ is also written as ${{A}^{4}}={{A}^{2}}\times {{A}^{2}}$. So we find the square matrix of A. After finding out the square matrix, we find out the ${{A}^{4}}$ and choose the correct option.

Complete step by step Solution:
We have given a matrix $A=\left( \begin{matrix}
   0 & 1 \\
   1 & 0 \\
\end{matrix} \right)$
Here matrix A is a $2\times 2$ matrix with 2 rows and 2 columns.
We have to find the value of ${{A}^{4}}$
We know ${{A}^{4}}={{A}^{2}}\times {{A}^{2}}$
To find ${{A}^{4}}$, first, we find the ${{A}^{2}}$.
We know ${{A}^{2}}=A\times A$
${{A}^{2}}=\left( \begin{matrix}
   0 & 1 \\
   1 & 0 \\
\end{matrix} \right)\left( \begin{matrix}
   0 & 1 \\
   1 & 0 \\
\end{matrix} \right)$
Now, we open the brackets and multiply the terms, we get
${{A}^{2}}=\left( \begin{matrix}
   0+1 & 0+0 \\
   0+0 & 1+0 \\
\end{matrix} \right)$
Adding the terms and simplifying them, we get
${{A}^{2}}=\left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right)$
Now we find out the value of ${{A}^{4}}$.
${{A}^{4}}$= $\left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right)\times \left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right)$
Again we open the brackets and multiply the terms, we get
${{A}^{4}}$= $\left( \begin{matrix}
   0+1 & 0+0 \\
   0+0 & 1+0 \\
\end{matrix} \right)$
Now we add the terms and get the new matrix which is
Then ${{A}^{4}}$= $\left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right)$
Hence the value of ${{A}^{4}}$= $\left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right)$

Therefore, the correct option is (A).

Note: Students must have the knowledge of solving the question related to matrix multiplication. By multiplication property, two matrices can be multiplied only when the columns of the first row are equal to the rows of the second matrix. Elements of a row of the first matrix will get multiplied with all the elements of the column of the second matrix and we will add them to get a new matrix. Students must do a lot of practice multiplying the two matrices otherwise they get confused while multiplying and choose the incorrect option.