If `A=[(1,1,1),(1,1,1),(1,1,1)]` then `A^(2)=`

Options

A. A
B. 2A
C. 3A
D. I

Answer

To find \( A^2 \) for the matrix \( A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \), we will perform matrix multiplication.

### Step 1: Write down the matrix \( A \)
We have:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \]

### Step 2: Calculate \( A^2 = A \times A \)
To find \( A^2 \), we multiply matrix \( A \) by itself. The element at position \( (i, j) \) in the resulting matrix is obtained by taking the dot product of the \( i \)-th row of the first matrix and the \( j \)-th column of the second matrix.

### Step 3: Calculate each element of \( A^2 \)

- **Element at (1,1)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (1,2)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (1,3)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (2,1)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (2,2)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (2,3)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (3,1)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (3,2)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

- **Element at (3,3)**:
\[ 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

### Step 4: Write the resulting matrix \( A^2 \)
Putting all the calculated elements together, we get:
\[ A^2 = \begin{pmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{pmatrix} \]

### Step 5: Conclusion
Thus, the final result is:
\[ A^2 = 3A \]
where \( A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \).