IDNLearn.com offers a unique blend of expert answers and community-driven knowledge. Explore a wide array of topics and find reliable answers from our experienced community members.
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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.