Connect with knowledgeable individuals and get your questions answered on IDNLearn.com. Ask your questions and receive prompt, detailed answers from our experienced and knowledgeable community members.

14. Given the following matrices, what is the correct product matrix [tex]\(AB\)[/tex]?

[tex]\( A = \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \)[/tex]

[tex]\( B = \left[\begin{array}{cc}-1 & 2 \\ 1 & 0\end{array}\right] \)[/tex]

A. [tex]\(\left[\begin{array}{lll}5 & 9 & 7\end{array}\right]\)[/tex]

B. [tex]\(\left[\begin{array}{cc}2 & -4 \\ 4 & 0 \\ 1 & 7\end{array}\right]\)[/tex]

C. [tex]\(\left[\begin{array}{ll}1 & 2 \\ 1 & 6\end{array}\right]\)[/tex]

D. [tex]\(\left[\begin{array}{ll}5 & 6 \\ 1 & 2\end{array}\right]\)[/tex]

E. [tex]\(\left[\begin{array}{ll}0 & 4 \\ 4 & 4\end{array}\right]\)[/tex]

F. [tex]\(\left[\begin{array}{cc}-1 & 4 \\ 3 & 0\end{array}\right]\)[/tex]


Sagot :

To determine the product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to perform matrix multiplication. Given the matrices:

[tex]\[ A = \left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right] \quad \text{and} \quad B = \left[\begin{array}{cc}-1 & 2 \\ 1 & 0\end{array}\right], \][/tex]

we will calculate the product matrix [tex]\( AB = C \)[/tex].

Matrix [tex]\( C \)[/tex] will also be a [tex]\( 2 \times 2 \)[/tex] matrix, and to find each entry [tex]\( c_{ij} \)[/tex] of matrix [tex]\( C \)[/tex], we use the dot product of the [tex]\( i \)[/tex]-th row of [tex]\( A \)[/tex] with the [tex]\( j \)[/tex]-th column of [tex]\( B \)[/tex].

1. For the entry [tex]\( c_{11} \)[/tex] (first row, first column):
[tex]\[ c_{11} = (1 \cdot -1) + (2 \cdot 1) = -1 + 2 = 1 \][/tex]

2. For the entry [tex]\( c_{12} \)[/tex] (first row, second column):
[tex]\[ c_{12} = (1 \cdot 2) + (2 \cdot 0) = 2 + 0 = 2 \][/tex]

3. For the entry [tex]\( c_{21} \)[/tex] (second row, first column):
[tex]\[ c_{21} = (3 \cdot -1) + (4 \cdot 1) = -3 + 4 = 1 \][/tex]

4. For the entry [tex]\( c_{22} \)[/tex] (second row, second column):
[tex]\[ c_{22} = (3 \cdot 2) + (4 \cdot 0) = 6 + 0 = 6 \][/tex]

Thus, the product matrix [tex]\( AB = C \)[/tex] is:

[tex]\[ C = \left[\begin{array}{cc}1 & 2 \\ 1 & 6\end{array}\right] \][/tex]

Therefore, the correct product matrix is:
[tex]\[ \left[\begin{array}{cc}1 & 2 \\ 1 & 6\end{array}\right] \][/tex]

So, the correct answer is option D:

[tex]\[ \left[\begin{array}{ll}1 & 2 \\ 1 & 6\end{array}\right] \][/tex]