Step 1: Matrix dimensions and compatibility
First, we note down the dimensions of the given matrices and verify the compatibility for multiplication.
Matrix A (1x4): \(\left[\begin{array}{llll}1 & -1 & 2 & 3\end{array}\right]\)
Matrix B (4x3): \(\left[\begin{array}{rrr}-1 & 2 & 0 \\\ 2 & -1 & 0 \\\ 0 & 5 & 2 \\\ -1 & 8 & 1\end{array}\right]\)
Since the number of columns of Matrix A (4) is equal to the number of rows of Matrix B (4), they are compatible for multiplication. The resulting matrix will have dimensions 1x3.
Step 2: Multiply the matrices by hand
We will now multiply the matrices element by element.
For the resulting matrix, C (1x3):
C[1, 1] = A[1, 1] * B[1, 1] + A[1, 2] * B[2, 1] + A[1, 3] * B[3, 1] + A[1, 4] * B[4, 1]
C[1, 1] = (1 * -1) + (-1 * 2) + (2 * 0) + (3 * -1) = -1 - 2 - 3 = -6
C[1, 2] = A[1, 1] * B[1, 2] + A[1, 2] * B[2, 2] + A[1, 3] * B[3, 2] + A[1, 4] * B[4, 2]
C[1, 2] = (1 * 2) + (-1 * -1) + (2 * 5) + (3 * 8) = 2 + 1 + 10 + 24 = 37
C[1, 3] = A[1, 1] * B[1, 3] + A[1, 2] * B[2, 3] + A[1, 3] * B[3, 3] + A[1, 4] * B[4, 3]
C[1, 3] = (1 * 0) + (-1 * 0) + (2 * 2) + (3 * 1) = 0 + 0 + 4 + 3 = 7
The resulting matrix C (1x3) is: \(\left[\begin{array}{lll}-6 & 37 & 7\end{array}\right]\)
Step 3: Multiply the matrices using technology
We can also use technology (such as calculator, Python, or MATLAB) to multiply the given matrices. Here, we'll use Python code as an example:
```python
import numpy as np
A = np.array([[1, -1, 2, 3]])
B = np.array([[-1, 2, 0], [2, -1, 0], [0, 5, 2], [-1, 8, 1]])
C = np.dot(A, B)
print(C)
```
The output will be: `[[-6, 37, 7]]`
Both by hand and by using technology, we get the same resulting matrix: \(\left[\begin{array}{lll}-6 & 37 & 7\end{array}\right]\)
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