If matrix `A=[a_(ij)]_(2x2),` where `a_(ij)={{:(1"," , i ne j),(0",", i=j):}` then `A^(3)` is equal to

Options

A. A
B. I
C. O
D. none of these

Answer

To solve the problem, we need to find \( A^3 \) for the matrix \( A \) defined as follows:

\[ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

### Step 1: Calculate \( A^2 \)

First, we need to calculate \( A^2 \) by multiplying matrix \( A \) with itself:

\[ A^2 = A \times A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \times \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

Using the rule of matrix multiplication, we compute each entry:

- First row, first column:
\[ 0 \times 0 + 1 \times 1 = 0 + 1 = 1 \]
- First row, second column:
\[ 0 \times 1 + 1 \times 0 = 0 + 0 = 0 \]
- Second row, first column:
\[ 1 \times 0 + 0 \times 1 = 0 + 0 = 0 \]
- Second row, second column:
\[ 1 \times 1 + 0 \times 0 = 1 + 0 = 1 \]

Thus, we have:
\[ A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 2: Calculate \( A^3 \)

Next, we calculate \( A^3 \) by multiplying \( A \) with \( A^2 \):

\[ A^3 = A \times A^2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Again, we compute each entry:

- First row, first column:
\[ 0 \times 1 + 1 \times 0 = 0 + 0 = 0 \]
- First row, second column:
\[ 0 \times 0 + 1 \times 1 = 0 + 1 = 1 \]
- Second row, first column:
\[ 1 \times 1 + 0 \times 0 = 1 + 0 = 1 \]
- Second row, second column:
\[ 1 \times 0 + 0 \times 1 = 0 + 0 = 0 \]

Thus, we have:
\[ A^3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

### Conclusion

We find that:
\[ A^3 = A \]