To find the product of the matrices [tex]\( X \)[/tex] and [tex]\( Y \)[/tex], we first need to understand their dimensions and the rules for matrix multiplication. Matrix [tex]\(X\)[/tex] is a [tex]\(1 \times 2\)[/tex] matrix and matrix [tex]\(Y\)[/tex] is a [tex]\(2 \times 1\)[/tex] matrix.
Given:
[tex]\[ X = \begin{bmatrix} -1 & 0 \end{bmatrix} \][/tex]
[tex]\[ Y = \begin{bmatrix} -2 \\ -1 \end{bmatrix} \][/tex]
When multiplying a [tex]\(1 \times 2\)[/tex] matrix [tex]\(X\)[/tex] by a [tex]\(2 \times 1\)[/tex] matrix [tex]\(Y\)[/tex], the resulting matrix will be a [tex]\(1 \times 1\)[/tex] matrix.
The multiplication is performed as follows:
[tex]\[ X \times Y = \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} -2 \\ -1 \end{bmatrix} \][/tex]
[tex]\[ = (-1 \times -2) + (0 \times -1) \][/tex]
Now, we compute each term:
- [tex]\((-1) \times (-2) = 2\)[/tex]
- [tex]\(0 \times (-1) = 0\)[/tex]
Adding these results together:
[tex]\[ 2 + 0 = 2 \][/tex]
Thus, the product [tex]\(XY\)[/tex] is:
[tex]\[ XY = \begin{bmatrix} 2 \end{bmatrix} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
