Evaluate the following : `[[2],[4],[6]][[1,2,3]]`

Answer

To evaluate the matrix multiplication of `[[2],[4],[6]]` and `[[1,2,3]]`, we will follow the steps of matrix multiplication.

### Step-by-Step Solution:

1. **Identify the Matrices and Their Orders**:
- The first matrix is `A = [[2], [4], [6]]`, which has 3 rows and 1 column. Thus, its order is **3 x 1**.
- The second matrix is `B = [[1, 2, 3]]`, which has 1 row and 3 columns. Thus, its order is **1 x 3**.

2. **Check Compatibility for Multiplication**:
- For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, matrix A has 1 column and matrix B has 1 row, so they can be multiplied.

3. **Determine the Order of the Resultant Matrix**:
- The resultant matrix will have the number of rows from the first matrix and the number of columns from the second matrix. Therefore, the resultant matrix will be of order **3 x 3**.

4. **Perform the Multiplication**:
- We will multiply each element of the first matrix by each element of the second matrix. The entries of the resultant matrix will be calculated as follows:

- **First Row**:
- (2 * 1) = 2
- (2 * 2) = 4
- (2 * 3) = 6
- Thus, the first row of the resultant matrix is `[2, 4, 6]`.

- **Second Row**:
- (4 * 1) = 4
- (4 * 2) = 8
- (4 * 3) = 12
- Thus, the second row of the resultant matrix is `[4, 8, 12]`.

- **Third Row**:
- (6 * 1) = 6
- (6 * 2) = 12
- (6 * 3) = 18
- Thus, the third row of the resultant matrix is `[6, 12, 18]`.

5. **Combine the Rows to Form the Resultant Matrix**:
- The final resultant matrix after multiplication is:
\[ \begin{bmatrix} 2 & 4 & 6 \\ 4 & 8 & 12 \\ 6 & 12 & 18 \end{bmatrix} \]

### Final Result:
The resultant matrix is:
\[ \begin{bmatrix} 2 & 4 & 6 \\ 4 & 8 & 12 \\ 6 & 12 & 18 \end{bmatrix} \]