When performing matrix multiplication, it's important to ensure that the number of columns in the first matrix matches the number of rows in the second matrix. Additionally, the dimensions of the resulting product matrix are determined by the number of rows from the first matrix and the number of columns from the second matrix. Make sure to check these conditions before proceeding with the multiplication.
Consider matrices \( [A]_{3 \times 2} \), \( [B]_{x \times y} \), and \( [C]_{3 \times 1} \).
For the product \( [A][B] \) to be defined, the number of columns in \(A\) (which is 2) must equal the number of rows in \(B\). Therefore, \( x = 2 \).
The dimensions of the resulting product \( [A][B] \) will be \( 3 \times y \). For this product to be equal to \( [C]_{3 \times 1} \), the dimensions must match. Thus, \( y = 1 \).
Consequently, the values are \( x = 2 \) and \( y = 1 \).
The correct option is:
\[ x = 2, \, y = 1. \]