Browse Questions

# Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.

Toolbox:
• If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
• A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j.
Given
$A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$
Transpose of a matrix can be obtained by interchanging the rows and columns.
$A'=\begin{bmatrix}1 & -1 & 5\\-1 & 2 & 1\\5 & 1 & 3\end{bmatrix}$
A square matrix is said to be symmetric if A'=A.
$\Rightarrow [a_{ji}]=[a_{ij}]$ for all values of i & j.
$a_{21}=-1=a_{12}$
$a_{31}=5=a_{13}$
$a_{32}=1=a_{23}$
$a_{11}=1=a_{11}$
$a_{22}=2=a_{22}$
$a_{33}=3=a_{33}$
Hence $a_{ji}=a_{ij}$
Therefore A is symmetric matrix.