To multiply the given matrices [tex]\(\left[\begin{array}{rr}5 & 0 \\ 3 & -5\end{array}\right]\)[/tex] and [tex]\(\left[\begin{array}{ll}2 & -1 \\ 2 & -2\end{array}\right]\)[/tex], we follow the standard procedure of matrix multiplication, where each element in the resulting matrix is obtained by taking the dot product of the corresponding row of the first matrix with the corresponding column of the second matrix.
Let's denote the matrices as:
[tex]\[ A = \left[\begin{array}{rr}5 & 0 \\ 3 & -5\end{array}\right] \][/tex]
[tex]\[ B = \left[\begin{array}{ll}2 & -1 \\ 2 & -2\end{array}\right] \][/tex]
We will calculate each element of the resulting matrix [tex]\( C = AB \)[/tex]. The resulting matrix [tex]\( C \)[/tex] will also be a [tex]\(2 \times 2\)[/tex] matrix.
1. First, calculate the element in the first row and first column of [tex]\( C \)[/tex] (denoted as [tex]\( C_{11} \)[/tex]):
[tex]\[ C_{11} = 5 \cdot 2 + 0 \cdot 2 = 10 + 0 = 10 \][/tex]
2. Next, calculate the element in the first row and second column of [tex]\( C \)[/tex] (denoted as [tex]\( C_{12} \)[/tex]):
[tex]\[ C_{12} = 5 \cdot (-1) + 0 \cdot (-2) = -5 + 0 = -5 \][/tex]
3. Now, calculate the element in the second row and first column of [tex]\( C \)[/tex] (denoted as [tex]\( C_{21} \)[/tex]):
[tex]\[ C_{21} = 3 \cdot 2 + (-5) \cdot 2 = 6 - 10 = -4 \][/tex]
4. Finally, calculate the element in the second row and second column of [tex]\( C \)[/tex] (denoted as [tex]\( C_{22} \)[/tex]):
[tex]\[ C_{22} = 3 \cdot (-1) + (-5) \cdot (-2) = -3 + 10 = 7 \][/tex]
Putting these together, the resulting matrix [tex]\( C \)[/tex] is:
[tex]\[ C = \left[\begin{array}{rr}10 & -5 \\ -4 & 7\end{array}\right] \][/tex]
Therefore, the correct answer is:
C. [tex]\(\left[\begin{array}{rr}10 & -5 \\ -4 & 7\end{array}\right]\)[/tex]
