Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve for [tex]\( A - B \)[/tex] given that [tex]\( \left[\begin{array}{ccc}1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4\end{array}\right]=A+B \)[/tex] where [tex]\( A \)[/tex] is a symmetric matrix and [tex]\( B \)[/tex] is a skew-symmetric matrix, we follow these steps:
1. Understand the properties of symmetric and skew-symmetric matrices:
- A symmetric matrix [tex]\( A \)[/tex] satisfies [tex]\( A^T = A \)[/tex].
- A skew-symmetric matrix [tex]\( B \)[/tex] satisfies [tex]\( B^T = -B \)[/tex] and the diagonal elements of [tex]\( B \)[/tex] are zero.
2. Formulate the given matrix [tex]\( A + B \)[/tex]:
[tex]\[ M = \left[\begin{array}{ccc} 1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4 \end{array}\right] \][/tex]
3. Derive the symmetric matrix [tex]\( A \)[/tex]:
Since [tex]\( A \)[/tex] is symmetric, we can use the property of symmetric matrices [tex]\( A = \frac{1}{2}(M + M^T) \)[/tex] to find [tex]\( A \)[/tex]. Calculating the transpose of [tex]\( M \)[/tex] and then averaging:
[tex]\[ M^T = \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \][/tex]
[tex]\[ A = \frac{1}{2} \left( \left[\begin{array}{ccc} 1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4 \end{array}\right] + \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \right) = \frac{1}{2} \left[\begin{array}{ccc} 2 & 4 & -4 \\ 4 & 0 & -6 \\ -4 & -6 & 8 \end{array}\right] = \left[\begin{array}{ccc} 1 & 2 & -2 \\ 2 & 0 & -3 \\ -2 & -3 & 4 \end{array}\right] \][/tex]
4. Derive the skew-symmetric matrix [tex]\( B \)[/tex]:
Since [tex]\( B \)[/tex] is skew-symmetric, we can use the property of skew-symmetric matrices [tex]\( B = \frac{1}{2}(M - M^T) \)[/tex] to find [tex]\( B \)[/tex]:
[tex]\[ B = \frac{1}{2} \left( \left[\begin{array}{ccc} 1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4 \end{array}\right] - \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \right) = \frac{1}{2} \left[\begin{array}{ccc} 0 & 2 & 4 \\ -2 & 0 & 2 \\ -4 & -2 & 0 \end{array}\right] = \left[\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 0 & 1 \\ -2 & -1 & 0 \end{array}\right] \][/tex]
5. Compute [tex]\( A - B \)[/tex]:
Finally, we subtract [tex]\( B \)[/tex] from [tex]\( A \)[/tex]:
[tex]\[ A - B = \left[\begin{array}{ccc} 1 & 2 & -2 \\ 2 & 0 & -3 \\ -2 & -3 & 4 \end{array}\right] - \left[\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 0 & 1 \\ -2 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \][/tex]
Thus, the matrix [tex]\( A - B \)[/tex] is:
[tex]\[ \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \][/tex]
1. Understand the properties of symmetric and skew-symmetric matrices:
- A symmetric matrix [tex]\( A \)[/tex] satisfies [tex]\( A^T = A \)[/tex].
- A skew-symmetric matrix [tex]\( B \)[/tex] satisfies [tex]\( B^T = -B \)[/tex] and the diagonal elements of [tex]\( B \)[/tex] are zero.
2. Formulate the given matrix [tex]\( A + B \)[/tex]:
[tex]\[ M = \left[\begin{array}{ccc} 1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4 \end{array}\right] \][/tex]
3. Derive the symmetric matrix [tex]\( A \)[/tex]:
Since [tex]\( A \)[/tex] is symmetric, we can use the property of symmetric matrices [tex]\( A = \frac{1}{2}(M + M^T) \)[/tex] to find [tex]\( A \)[/tex]. Calculating the transpose of [tex]\( M \)[/tex] and then averaging:
[tex]\[ M^T = \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \][/tex]
[tex]\[ A = \frac{1}{2} \left( \left[\begin{array}{ccc} 1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4 \end{array}\right] + \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \right) = \frac{1}{2} \left[\begin{array}{ccc} 2 & 4 & -4 \\ 4 & 0 & -6 \\ -4 & -6 & 8 \end{array}\right] = \left[\begin{array}{ccc} 1 & 2 & -2 \\ 2 & 0 & -3 \\ -2 & -3 & 4 \end{array}\right] \][/tex]
4. Derive the skew-symmetric matrix [tex]\( B \)[/tex]:
Since [tex]\( B \)[/tex] is skew-symmetric, we can use the property of skew-symmetric matrices [tex]\( B = \frac{1}{2}(M - M^T) \)[/tex] to find [tex]\( B \)[/tex]:
[tex]\[ B = \frac{1}{2} \left( \left[\begin{array}{ccc} 1 & 3 & 0 \\ 1 & 0 & -2 \\ -4 & -4 & 4 \end{array}\right] - \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \right) = \frac{1}{2} \left[\begin{array}{ccc} 0 & 2 & 4 \\ -2 & 0 & 2 \\ -4 & -2 & 0 \end{array}\right] = \left[\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 0 & 1 \\ -2 & -1 & 0 \end{array}\right] \][/tex]
5. Compute [tex]\( A - B \)[/tex]:
Finally, we subtract [tex]\( B \)[/tex] from [tex]\( A \)[/tex]:
[tex]\[ A - B = \left[\begin{array}{ccc} 1 & 2 & -2 \\ 2 & 0 & -3 \\ -2 & -3 & 4 \end{array}\right] - \left[\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 0 & 1 \\ -2 & -1 & 0 \end{array}\right] = \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \][/tex]
Thus, the matrix [tex]\( A - B \)[/tex] is:
[tex]\[ \left[\begin{array}{ccc} 1 & 1 & -4 \\ 3 & 0 & -4 \\ 0 & -2 & 4 \end{array}\right] \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
