# What is the matrix product \( \begin{bmatrix} a \\ 2a \\ 3a \end{bmatrix} \begin{bmatrix} 1 & 0 & -1 \end{bmatrix} \)

What is the matrix product \( \begin{bmatrix} a \\ 2a \\ 3a \end{bmatrix} \begin{bmatrix} 1 & 0 & -1 \end{bmatrix} \) ?

- \( \begin{bmatrix} a & 0 & -a \\ 2a & 0 & -2a \\ 3a & 0 & -3a \end{bmatrix} \)
- \( \begin{bmatrix} a & 2a & 3a \\ 0 & 0 & 0 \\ -a & -2a & -3a \end{bmatrix} \)
- \( \begin{bmatrix} 2a & 0 & -2a \end{bmatrix} \)
- \( \begin{bmatrix} 6a & 0 & -6a \end{bmatrix} \)
- \( \begin{bmatrix} 0 \end{bmatrix} \)

### Answer

**The correct answer is A. **

To calculate a matrix product, you go across each row in the first matrix and down each column in the second matrix. You multiply the terms from a row by the corresponding terms from the column, and you add all those terms together for the row-column combination and put the sum in that row-column of the result matrix. You do this for each row-column combination.

In this case, there is only 1 element in each row of the first matrix, and 1 element in each column in the second matrix.

First row = (a x 1), (a x 0), (a x -1)

Second row = (2a x 1), (2a x 0), (2a x -1)

Third row = First row = (3a x 1), (3a x 0), (3a x -1)