Certainly! Let's solve the given matrix multiplication step-by-step.
We are given two matrices:
[tex]\[ A = \begin{pmatrix} -2 & 5 \\ 1 & -2 \end{pmatrix} \][/tex]
and
[tex]\[ B = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \][/tex]
We are asked to find the product of matrix [tex]\( A \)[/tex] and vector [tex]\( B \)[/tex]. The product of these matrices can be represented as:
[tex]\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -2 & 5 \\ 1 & -2 \end{pmatrix} \begin{pmatrix} -1 \\ 2 \end{pmatrix} \][/tex]
To find the resulting vector, we perform the following matrix multiplication:
1. Calculate the first element [tex]\( x \)[/tex] in the resulting vector:
[tex]\[ x = (-2 \times -1) + (5 \times 2) \][/tex]
[tex]\[ x = 2 + 10 \][/tex]
[tex]\[ x = 12 \][/tex]
2. Calculate the second element [tex]\( y \)[/tex] in the resulting vector:
[tex]\[ y = (1 \times -1) + (-2 \times 2) \][/tex]
[tex]\[ y = -1 - 4 \][/tex]
[tex]\[ y = -5 \][/tex]
Therefore, the resulting vector is:
[tex]\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 12 \\ -5 \end{pmatrix} \][/tex]
So, the final result is:
[tex]\[ \begin{pmatrix} 12 \\ -5 \end{pmatrix} \][/tex]
Thus, [tex]\( x = 12 \)[/tex] and [tex]\( y = -5 \)[/tex].
