Math Problem Statement
Solve ii. Step by step .
Solution
Let's solve part (ii) of the problem step by step:
We are given matrix and asked to show that is a symmetric matrix.
Step 1: Recall the Definition of Symmetric Matrix
A matrix is symmetric if . That is, a matrix is symmetric if it is equal to its own transpose.
Step 2: Write Down the Matrix
The given matrix is:
1 & 3 & -1 \\ -3 & -5 & 8 \\ 1 & 8 & 9 \end{pmatrix}$$ ### Step 3: Compute the Transpose of $$A$$, Denoted by $$A^T$$ To find the transpose $$A^T$$, we swap the rows and columns of $$A$$. The result is: $$A^T = \begin{pmatrix} 1 & -3 & 1 \\ 3 & -5 & 8 \\ -1 & 8 & 9 \end{pmatrix}$$ ### Step 4: Add $$A$$ and $$A^T$$ Now, we compute $$A + A^T$$. This is done by adding the corresponding elements of $$A$$ and $$A^T$$. $$A + A^T = \begin{pmatrix} 1 & 3 & -1 \\ -3 & -5 & 8 \\ 1 & 8 & 9 \end{pmatrix} + \begin{pmatrix} 1 & -3 & 1 \\ 3 & -5 & 8 \\ -1 & 8 & 9 \end{pmatrix}$$ Perform the element-wise addition: $$A + A^T = \begin{pmatrix} 1 + 1 & 3 + (-3) & -1 + 1 \\ -3 + 3 & -5 + (-5) & 8 + 8 \\ 1 + (-1) & 8 + 8 & 9 + 9 \end{pmatrix}$$ This simplifies to: $$A + A^T = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -10 & 16 \\ 0 & 16 & 18 \end{pmatrix}$$ ### Step 5: Check if $$A + A^T$$ is Symmetric For $$A + A^T$$ to be symmetric, it must satisfy the condition $$(A + A^T) = (A + A^T)^T$$. In other words, the matrix must be equal to its transpose. Let's take the transpose of $$A + A^T$$: $$(A + A^T)^T = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -10 & 16 \\ 0 & 16 & 18 \end{pmatrix}^T$$ Since the transpose of a symmetric matrix is itself, we find: $$(A + A^T)^T = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -10 & 16 \\ 0 & 16 & 18 \end{pmatrix}$$ Clearly, $$A + A^T = (A + A^T)^T$$, confirming that $$A + A^T$$ is indeed symmetric. ### Final Answer: Yes, $$A + A^T$$ is a symmetric matrix. --- Do you want details or have any questions? Here are 5 relative questions for more practice: 1. What is the condition for a matrix to be skew-symmetric? 2. How do you compute the determinant of a 3x3 matrix? 3. If $$A$$ is a symmetric matrix, what can be said about its eigenvalues? 4. Can a non-square matrix be symmetric or skew-symmetric? 5. How is matrix multiplication affected when using symmetric matrices? **Tip**: To check if a matrix is symmetric or skew-symmetric, always compute its transpose and compare it to the original matrix!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Symmetry in Matrices
Matrix Transpose
Formulas
Transpose of a matrix A^T: Swap rows and columns of A
Symmetric matrix condition: A = A^T
Theorems
A matrix is symmetric if it equals its transpose (A = A^T)
Suitable Grade Level
Grades 10-12 or introductory college level
Related Recommendation
Matrix Algebra: Proving Symmetry of a Given Matrix
Solve for 2x2 Matrices Satisfying AA^T = A^TA
Matrix Symmetry: Checking Symmetric and Skew-Symmetric Properties for a Given Matrix A
Understanding A^T A: Properties of Matrix Transpose and Multiplication
Prove that A^(-1) = A^T for a 3x3 Orthogonal Matrix