To find the product [tex]\( AB \)[/tex], where [tex]\( A \)[/tex] is a matrix and [tex]\( B \)[/tex] is a vector, we use matrix multiplication. Here is the detailed step-by-step process:
Given:
[tex]\[ A = \begin{pmatrix} 1 & 4 \\ 6 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \][/tex]
We need to calculate:
[tex]\[ AB = \begin{pmatrix} 1 & 4 \\ 6 & 3 \end{pmatrix} \begin{pmatrix} 2 \\ 5 \end{pmatrix} \][/tex]
Matrix multiplication involves taking the dot product of the rows of [tex]\( A \)[/tex] with the column vector [tex]\( B \)[/tex]. The resulting element in the [tex]\( i \)[/tex]-th row and [tex]\( j \)[/tex]-th column is calculated as:
1. For the first row of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ (1 \cdot 2) + (4 \cdot 5) = 2 + 20 = 22 \][/tex]
2. For the second row of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ (6 \cdot 2) + (3 \cdot 5) = 12 + 15 = 27 \][/tex]
Putting these together, the resultant vector [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{pmatrix} 22 \\ 27 \end{pmatrix} \][/tex]
Thus, the product [tex]\( AB \)[/tex] is:
[tex]\[ \boxed{\begin{pmatrix} 22 \\ 27 \end{pmatrix}} \][/tex]
