Problem 33 Should be done two ways: (a) by ... [FREE SOLUTION]

Chapter 3: Problem 33

Should be done two ways: (a) by hand; (b) using technology where possible. $$ A=\left[\begin{array}{rrrr} 0 & -1 & 0 & 1 \\ 10 & 0 & 1 & 0 \end{array}\right], B=\left[\begin{array}{rr} 0 & -1 \\ 1 & 1 \\ -1 & 3 \\ 5 & 0 \end{array}\right], C=\left[\begin{array}{rr} 1 & -1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{array}\right] $$ $A(B-C)$

Short Answer

Expert verified

In conclusion, the result of matrix multiplication A(B-C) is: \[ A(B-C) = \begin{bmatrix} 4 & -1 \\ -10 & 10 \end{bmatrix} \]

Step by step solution

(a) By hand: Find the difference between matrices B and C

First, we will find the difference B - C. To do this, we subtract the corresponding elements of the matrices: $ B - C = \begin{bmatrix} 0 & -1 \\ 1 & 1 \\ -1 & 3 \\ 5 & 0 \end{bmatrix} - \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix} $ Calculating the difference: $ D = B - C = \begin{bmatrix} -1 & 0 \\ 0 & 0 \\ -2 & 2 \\ 4 & -1 \end{bmatrix} $

(a) By hand: Multiply matrices A and D

Now, we will multiply matrix A with the difference matrix D (A*D): $ A * D = \begin{bmatrix} 0 & -1 & 0 & 1 \\ 10 & 0 & 1 & 0 \end{bmatrix} * \begin{bmatrix} -1 & 0 \\ 0 & 0 \\ -2 & 2 \\ 4 & -1 \end{bmatrix} $ Calculating the multiplication: $ A * D = \begin{bmatrix} 4 & -1 \\ -10 & 10 \end{bmatrix} $

(b) Using technology: Find the difference between matrices B and C

We can use technology, such as a calculator or software like MATLAB or NumPy, to find the difference between matrices B and C. In Python with NumPy, the code to find the difference would look as follows: ```python import numpy as np B = np.array([[0, -1], [1, 1], [-1, 3], [5, 0]]) C = np.array([[1, -1], [1, 1], [1, 1], [1, 1]]) D = np.subtract(B, C) print(D) ``` This should output the difference matrix D, which is the same as we found in the by hand method: \[ D = \begin{bmatrix} -1 & 0 \\ 0 & 0 \\ -2 & 2 \\ 4 & -1 \end{bmatrix} \]

(b) Using technology: Multiply matrices A and D

Using technology, we can also find the product of matrices A and D. For example, in Python with NumPy, the code would look like this: ```python A = np.array([[0, -1, 0, 1], [10, 0, 1, 0]]) result = np.matmul(A, D) print(result) ``` This should output the product matrix, which is the same as we found in the by hand method: \[ A * D = \begin{bmatrix} 4 & -1 \\ -10 & 10 \end{bmatrix} \] In conclusion, the result of matrix multiplication A(B-C) is: \[ A(B-C) = \begin{bmatrix} 4 & -1 \\ -10 & 10 \end{bmatrix} \]

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Step by step solution

(a) By hand: Find the difference between matrices B and C

(a) By hand: Multiply matrices A and D

(b) Using technology: Find the difference between matrices B and C

(b) Using technology: Multiply matrices A and D

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