To determine the product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to perform matrix multiplication.
Matrix [tex]\( A \)[/tex] has dimensions [tex]\( 2 \times 2 \)[/tex]:
[tex]\[ A = \begin{bmatrix} -1 & 4 \\ 0 & 5 \end{bmatrix} \][/tex]
Matrix [tex]\( B \)[/tex] has dimensions [tex]\( 2 \times 3 \)[/tex]:
[tex]\[ B = \begin{bmatrix} 7 & 9 & 2 \\ 2 & 0 & -3 \end{bmatrix} \][/tex]
The resulting matrix after multiplication [tex]\( AB \)[/tex] will have the dimensions [tex]\( 2 \times 3 \)[/tex] because the number of rows of [tex]\( A \)[/tex] is 2 and the number of columns of [tex]\( B \)[/tex] is 3.
We find each element of the product matrix [tex]\( AB \)[/tex] by taking the dot product of the corresponding row of [tex]\( A \)[/tex] with the column of [tex]\( B \)[/tex].
The element at position (1,1):
[tex]\[ (-1 \cdot 7) + (4 \cdot 2) = -7 + 8 = 1 \][/tex]
The element at position (1,2):
[tex]\[ (-1 \cdot 9) + (4 \cdot 0) = -9 + 0 = -9 \][/tex]
The element at position (1,3):
[tex]\[ (-1 \cdot 2) + (4 \cdot -3) = -2 + (-12) = -14 \][/tex]
The element at position (2,1):
[tex]\[ (0 \cdot 7) + (5 \cdot 2) = 0 + 10 = 10 \][/tex]
The element at position (2,2):
[tex]\[ (0 \cdot 9) + (5 \cdot 0) = 0 + 0 = 0 \][/tex]
The element at position (2,3):
[tex]\[ (0 \cdot 2) + (5 \cdot -3) = 0 + (-15) = -15 \][/tex]
Therefore, the product [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{bmatrix} 1 & -9 & -14 \\ 10 & 0 & -15 \end{bmatrix} \][/tex]
Comparing with the options, the correct answer is:
C. [tex]\(\left[\begin{array}{ccc}1 & -9 & -14 \\ 10 & 0 & -15\end{array}\right]\)[/tex]
