Solved Examples: Identifying and Proving Symmetric Matrices

📚 Quick Study Guide

• 🔢 Definition: A square matrix $A$ is symmetric if it is equal to its transpose, i.e., $A = A^T$.
• 🔄 Transpose: The transpose of a matrix $A$, denoted as $A^T$, is obtained by interchanging its rows and columns.
• 📍 Element-wise Condition: For a matrix $A = [a_{ij}]$, the condition for symmetry is $a_{ij} = a_{ji}$ for all $i$ and $j$.
• ✅ Proving Symmetry: To prove a matrix is symmetric, show that its transpose is equal to the original matrix.
• ➕ Symmetric Matrices and Operations: The sum and difference of symmetric matrices are also symmetric.
• 🧑‍🏫 Example: Consider the matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$. Its transpose $A^T = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$. Since $A = A^T$, the matrix $A$ is symmetric.

Practice Quiz

1. Question 1: Which of the following matrices is symmetric?
   1. A) $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$
   3. C) $\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$
2. Question 2: If $A$ is a symmetric matrix, then what is the relationship between $A$ and its transpose $A^T$?
   1. A) $A = -A^T$
   2. B) $A = A^T$
   3. C) $A = 2A^T$
   4. D) $A = \frac{1}{2}A^T$
3. Question 3: Given $A = \begin{bmatrix} x & 4 \\ 4 & y \end{bmatrix}$, for what values of $x$ and $y$ is $A$ symmetric?
   1. A) $x = 4, y = 4$
   2. B) $x = 1, y = 1$
   3. C) $x$ and $y$ can be any values.
   4. D) $x = 0, y = 0$
4. Question 4: Which of the following operations always results in a symmetric matrix if $A$ and $B$ are symmetric?
   1. A) $A + B$
   2. B) $A - B$
   3. C) $AB$
   4. D) $BA$
5. Question 5: Which of the following matrices is NOT symmetric?
   1. A) $\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$
   2. B) $\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$
   3. C) $\begin{bmatrix} 7 & 8 \\ -8 & 9 \end{bmatrix}$
   4. D) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
6. Question 6: If matrix $A$ is symmetric and $A^2 = I$ (Identity matrix), which of the following is true?
   1. A) $A = I$ only
   2. B) $A = -I$ only
   3. C) $A = I$ or $A = -I$
   4. D) $A$ must be orthogonal
7. Question 7: What condition must be met for a diagonal matrix to be symmetric?
   1. A) All diagonal elements must be equal to 1.
   2. B) All diagonal elements must be non-zero.
   3. C) It is always symmetric.
   4. D) All non-diagonal elements must be non-zero.