Types of Matrices Based on Symmetry
1. Symmetric Matrix
- A square matrix where the element at position (i, j) is equal to the element at position (j, i).
- In simpler terms, it is symmetrical about the diagonal.
- Example:
\[\Large
Symmetric\;Matrix\;A =
\begin{bmatrix}
\color{blue}{1} & \color{red}{2} & \color{green}{3} \\
\color{red}{2} & \color{blue}{4} & \color{orange}{5} \\
\color{green}{3} & \color{orange}{5} & \color{blue}{6}
\end{bmatrix}
\]
- \( A_{ij} = A_{ji} \)
- For example:
- \( A_{12} = A_{21} \)
- \( A_{13} = A_{31} \)
2. Skew-Symmetric Matrix
- A square matrix where the element at position (i, j) is the negative of the element at position (j, i), and diagonal elements are zero.
- In simpler terms, it is symmetrical with opposite signs.
- Example:
\[\Large
A =
\begin{bmatrix}
\color{blue}{0} & \color{red}{-2} & \color{green}{3} \\
\color{red}{2} & \color{blue}{0} & \color{orange}{-4} \\
\color{green}{-3} & \color{orange}{4} & \color{blue}{0}
\end{bmatrix}
\]
- \( A_{ij} = -A_{ji} \), meaning the element at \( (i, j) \) is the negative of the element at \( (j, i) \).
- For example:
- \( A_{12} = -A_{21} \), here \( A_{12} = -2 \) and \( A_{21} = 2 \).
- \( A_{13} = -A_{31} \), here \( A_{13} = 3 \) and \( A_{31} = -3 \).
- \( A_{23} = -A_{32} \), here \( A_{23} = -4 \) and \( A_{32} = 4 \).
Which of the following statements is TRUE about symmetric and skew-symmetric matrices?