[A]_3×2[B]_x×y = [C]_3×1, then x and y are:

The correct option is x = 2, y = 1.

To understand why this is correct, let us review the fundamental rule for the multiplication of two matrices.

For two matrices $A$ and $B$ to be multiplied, the number of columns in the first matrix $A$ must be equal to the number of rows in the second matrix $B$. Furthermore, if matrix $A$ has dimensions $m\times n$ and matrix $B$ has dimensions $n\times p$, the resulting matrix $C$ will have dimensions $m\times p$.

In this problem, we are given the matrix equation:
${\left[A\right]}_{3\times 2}{\left[B\right]}_{x\times y}={\left[C\right]}_{3\times 1}$

Let us apply the rules of matrix multiplication step-by-step:

1. Compatibility for Multiplication:
The number of columns in matrix $A$ is 2. The number of rows in matrix $B$ is $x$. For multiplication to be defined, these two values must be equal. Therefore:
$x=2$

2. Dimension of the Resulting Matrix:
The resulting matrix $C$ is formed by taking the number of rows of $A$ (which is 3) and the number of columns of $B$ (which is $y$). This means the dimensions of $C$ must be $3\times y$.
We are given that the dimensions of $C$ are $3\times 1$. By comparing the number of columns, we get:
$y=1$

Combining these two results, we find that $x=2$ and $y=1$, which corresponds exactly to the correct option.