1. The problem involves multiplying a 2x2 matrix $ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ by a vector $ \beta $. 2. To multiply a matrix by a vector, the vector must be compatible in dimensions. Assuming $ \beta = \begin{bmatrix} x \\ y \end{bmatrix} $, a 2x1 vector, the multiplication is valid. 3. The formula for matrix-vector multiplication is: $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} $$ 4. This means each element of the resulting vector is the dot product of the corresponding row of the matrix with the vector. 5. So the product is: $$ \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix} $$ This is the final answer for the multiplication of the matrix by vector $ \beta $.