To find the product of two matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], each entry in the resulting matrix is obtained by taking the dot product of the corresponding row from [tex]\( A \)[/tex] with the corresponding column from [tex]\( B \)[/tex]. Let's perform the multiplication step by step:
Given matrices:
[tex]\[ A = \left[\begin{array}{rr}-4 & 2 \\ 3 & 2\end{array}\right] \][/tex]
[tex]\[ B = \left[\begin{array}{rr}-2 & -1 \\ 5 & 1\end{array}\right] \][/tex]
We need to calculate entries for the resulting matrix [tex]\( AB \)[/tex], which is a 2x2 matrix.
1. First Row, First Column:
[tex]\[ (AB)_{1,1} = (-4 \times -2) + (2 \times 5) \][/tex]
[tex]\[ (AB)_{1,1} = 8 + 10 \][/tex]
[tex]\[ (AB)_{1,1} = 18 \][/tex]
2. First Row, Second Column:
[tex]\[ (AB)_{1,2} = (-4 \times -1) + (2 \times 1) \][/tex]
[tex]\[ (AB)_{1,2} = 4 + 2 \][/tex]
[tex]\[ (AB)_{1,2} = 6 \][/tex]
3. Second Row, First Column:
[tex]\[ (AB)_{2,1} = (3 \times -2) + (2 \times 5) \][/tex]
[tex]\[ (AB)_{2,1} = -6 + 10 \][/tex]
[tex]\[ (AB)_{2,1} = 4 \][/tex]
4. Second Row, Second Column:
[tex]\[ (AB)_{2,2} = (3 \times -1) + (2 \times 1) \][/tex]
[tex]\[ (AB)_{2,2} = -3 + 2 \][/tex]
[tex]\[ (AB)_{2,2} = -1 \][/tex]
Putting all these together, the resulting matrix [tex]\( AB \)[/tex] is:
[tex]\[ AB = \left[\begin{array}{rr}18 & 6 \\ 4 & -1\end{array}\right] \][/tex]
So, the correct answer is:
[tex]\[ \left[\begin{array}{rr}18 & 6 \\ 4 & -1\end{array}\right] \][/tex]
