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Problem 1

# Compute the products. Some of these may be undefined. Exercises marked $$\mathrm{T}$$ should be done by using technology. The others should be done in two ways: by hand and by using technology where possible. $\left[\begin{array}{lll}1 & 3 & -1\end{array}\right]\left[\begin{array}{r}9 \\\ 1 \\ -1\end{array}\right]$

Expert verified
The product of the given matrices is a 1x1 matrix with the single value, 13. This result was obtained by both manual multiplication and using technology (Python and NumPy).
See the step by step solution

## Step 1: Understand matrix multiplication

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix, and then summing the products. When multiplying a 1x3 matrix and a 3x1 matrix, the result will be a 1x1 matrix (a single number).

## Step 2: Multiply the matrices by hand

To multiply the given matrices by hand, we will multiply each element in the row of the first matrix by each corresponding element in the column of the second matrix, and then sum the results: $$\left[\begin{array}{lll}1\times9+3\times1+(-1)\times(-1)\end{array}\right] = \left[\begin{array}{lll}9+3+1\end{array}\right] = \left[\begin{array}{lll}13\end{array}\right]$$ So, the product of the given matrices is a 1x1 matrix with the single value, 13.

## Step 3: Multiply the matrices using technology

Matrix multiplication can be done using various technology tools, such as calculators, online tools, or programming languages like Python with the NumPy library. Below is an example using Python and NumPy: python import numpy as np A = np.array([[1, 3, -1]]) B = np.array([, , [-1]]) product = np.dot(A, B) print(product)  This code will output:  []  Which confirms our result from the manual multiplication method. The product of the given matrices is a 1x1 matrix with the single value, 13.

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