`A = ({:(1),(2),(3):}),"find " A * A^(T)`

Answer

To solve the problem of finding \( A \times A^T \) where \( A = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \), we will follow these steps:

### Step 1: Write down the matrix \( A \)
Given:
\[ A = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]

### Step 2: Find the transpose of matrix \( A \)
The transpose of a matrix is obtained by converting its rows into columns. Therefore, the transpose \( A^T \) is:
\[ A^T = \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} \]

### Step 3: Multiply \( A \) and \( A^T \)
Now we need to multiply \( A \) (which is a \( 3 \times 1 \) matrix) with \( A^T \) (which is a \( 1 \times 3 \) matrix). The result will be a \( 3 \times 3 \) matrix.

The multiplication is performed as follows:
\[ A \times A^T = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} \]

Calculating each element of the resulting matrix:
- First row:
- \( 1 \times 1 = 1 \)
- \( 1 \times 2 = 2 \)
- \( 1 \times 3 = 3 \)

- Second row:
- \( 2 \times 1 = 2 \)
- \( 2 \times 2 = 4 \)
- \( 2 \times 3 = 6 \)

- Third row:
- \( 3 \times 1 = 3 \)
- \( 3 \times 2 = 6 \)
- \( 3 \times 3 = 9 \)

Putting it all together, we get:
\[ A \times A^T = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \]

### Final Answer
Thus, the result of \( A \times A^T \) is:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \]