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To solve the problem of multiplying the given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], follow these steps:
1. Understand the Dimensions:
The first matrix [tex]\( A \)[/tex] is a 2x2 matrix:
[tex]\[ A = \begin{pmatrix} -1 & 4 \\ 0 & 5 \end{pmatrix} \][/tex]
The second matrix [tex]\( B \)[/tex] is a 2x3 matrix:
[tex]\[ B = \begin{pmatrix} 7 & 9 & 2 \\ 2 & 0 & -3 \end{pmatrix} \][/tex]
2. Check if the matrices can be multiplied:
For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, [tex]\( A \)[/tex] has 2 columns and [tex]\( B \)[/tex] has 2 rows, so matrix multiplication is possible.
3. Multiply the matrices:
The product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( AB \)[/tex], will be a matrix whose elements are computed as follows:
[tex]\[ AB_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj} \][/tex]
For each element in the resulting matrix, perform the dot product of the corresponding row from [tex]\( A \)[/tex] with the corresponding column from [tex]\( B \)[/tex].
Calculate each element of the resulting matrix [tex]\( AB \)[/tex]:
- [tex]\( \text{Element } (1,1): \)[/tex]
[tex]\[ (-1 \cdot 7) + (4 \cdot 2) = -7 + 8 = 1 \][/tex]
- [tex]\( \text{Element } (1,2): \)[/tex]
[tex]\[ (-1 \cdot 9) + (4 \cdot 0) = -9 + 0 = -9 \][/tex]
- [tex]\( \text{Element } (1,3): \)[/tex]
[tex]\[ (-1 \cdot 2) + (4 \cdot -3) = -2 - 12 = -14 \][/tex]
- [tex]\( \text{Element } (2,1): \)[/tex]
[tex]\[ (0 \cdot 7) + (5 \cdot 2) = 0 + 10 = 10 \][/tex]
- [tex]\( \text{Element } (2,2): \)[/tex]
[tex]\[ (0 \cdot 9) + (5 \cdot 0) = 0 + 0 = 0 \][/tex]
- [tex]\( \text{Element } (2,3): \)[/tex]
[tex]\[ (0 \cdot 2) + (5 \cdot -3) = 0 - 15 = -15 \][/tex]
4. Form the Resulting Matrix:
After performing the calculations, the resulting matrix [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{pmatrix} 1 & -9 & -14 \\ 10 & 0 & -15 \end{pmatrix} \][/tex]
5. Select the Correct Answer:
Among the options provided, the correct one is:
[tex]\[ \text{D.}\ \left[\begin{array}{ccc} 1 & -9 & -14 \\ 10 & 0 & -15 \end{array}\right] \][/tex]
Therefore, the product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \boxed{D} \)[/tex].
1. Understand the Dimensions:
The first matrix [tex]\( A \)[/tex] is a 2x2 matrix:
[tex]\[ A = \begin{pmatrix} -1 & 4 \\ 0 & 5 \end{pmatrix} \][/tex]
The second matrix [tex]\( B \)[/tex] is a 2x3 matrix:
[tex]\[ B = \begin{pmatrix} 7 & 9 & 2 \\ 2 & 0 & -3 \end{pmatrix} \][/tex]
2. Check if the matrices can be multiplied:
For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, [tex]\( A \)[/tex] has 2 columns and [tex]\( B \)[/tex] has 2 rows, so matrix multiplication is possible.
3. Multiply the matrices:
The product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( AB \)[/tex], will be a matrix whose elements are computed as follows:
[tex]\[ AB_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj} \][/tex]
For each element in the resulting matrix, perform the dot product of the corresponding row from [tex]\( A \)[/tex] with the corresponding column from [tex]\( B \)[/tex].
Calculate each element of the resulting matrix [tex]\( AB \)[/tex]:
- [tex]\( \text{Element } (1,1): \)[/tex]
[tex]\[ (-1 \cdot 7) + (4 \cdot 2) = -7 + 8 = 1 \][/tex]
- [tex]\( \text{Element } (1,2): \)[/tex]
[tex]\[ (-1 \cdot 9) + (4 \cdot 0) = -9 + 0 = -9 \][/tex]
- [tex]\( \text{Element } (1,3): \)[/tex]
[tex]\[ (-1 \cdot 2) + (4 \cdot -3) = -2 - 12 = -14 \][/tex]
- [tex]\( \text{Element } (2,1): \)[/tex]
[tex]\[ (0 \cdot 7) + (5 \cdot 2) = 0 + 10 = 10 \][/tex]
- [tex]\( \text{Element } (2,2): \)[/tex]
[tex]\[ (0 \cdot 9) + (5 \cdot 0) = 0 + 0 = 0 \][/tex]
- [tex]\( \text{Element } (2,3): \)[/tex]
[tex]\[ (0 \cdot 2) + (5 \cdot -3) = 0 - 15 = -15 \][/tex]
4. Form the Resulting Matrix:
After performing the calculations, the resulting matrix [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{pmatrix} 1 & -9 & -14 \\ 10 & 0 & -15 \end{pmatrix} \][/tex]
5. Select the Correct Answer:
Among the options provided, the correct one is:
[tex]\[ \text{D.}\ \left[\begin{array}{ccc} 1 & -9 & -14 \\ 10 & 0 & -15 \end{array}\right] \][/tex]
Therefore, the product of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \boxed{D} \)[/tex].
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