Class notes mathematics matrix BUSINESS MATHEMATICS., ISBN: 9789351426806

Symmetric Matrix
A square Matrix A is said to be symmetric if aij=ajifor all i and j, whereaij is an
element present at (i,j)th position (ith row and jth column in matrix A) and aji is
an element present at (j,i)th position (jth row and ith column in matrix A).

In other words, we can say that matrix A is said to be symmetric if
transpose of matrix A is equal to matrix A itself (AT=A).
A=

1 2 5
2 5 -7
5 -7 3

It is symmetric matrix becauseaij=ajifor all i and j
Example, a12=a21=2,a13=a31=5 anda23=a32=−7

In other words, transpose of Matrix A is equal to matrix Aitself (AT=A) which means matrix A is
symmetric.


Skew-Symmetric Matrix
Square matrix A is said to be skew-symmetric if aij=−aji for all i and j. In other words, we can say
that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A
i.e (AT=−A). Note that all the main diagonal elements in the skew-symmetric matrix are zero.


0 -5 4
5 0 -1
-4 1 0

, Above is skew-symmetric matrix because aij=−aji for all i and j and leading diagonal elements are
zero.



A= -AT and leading diagonal elements are
zero.


Singular and Non singular Matrix
Non Singular Matrix
A square matrix A is said to be non-singular if |A| = det (A) ≠ 0