Properties of Skew Symmetric Matrices linear algebra csir net gate iit jam mathematics msc entrance

For Notes and Practice set WhatsApp @ 8130648819 or visit our Website

Join this channel to get access to perks:

Real Analysis: Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets, completeness of R. Power series (of real variable), Taylor's series, radius and interval of convergence, term-wise differentiation and integration of power series.

Jam 2019 MCQ MA Real Analysis University Entrance Exam DU

Properties of Skew-Symmetric Matrix
(i) In a Skew-symmetric matrix the elements below the principal diagonal are negatively equaly to the elements
above the principal diagonal.
(ii) All the elements in the principal diagonal of a Skew-symmetric matrix are equal to zero,i.e.,a_ii=0.
Exp.Let A=[a_ij ]_(m×m) be a Skew-symmetric matrix.Then,
a_ij=-a_ji
Let j=i,
∴ a_(ii )=-a_ii
⇒ 2a_ii=0
⇒ a_ii=0
(iii) Every square null matrix is a Skew-symmetric matrix but none of diagonal matrix,scalar matrix and identity
matrix are Skew-symmteric matrix.
(iv) Every Skew-symmetric matrix is necessarily be square matrix.
(v) If A is Skew-symmetric matrix,then kA is also Skew-symmetric matrix for any scalar k.
(vi) If A and B are Skew-symmetric matrices of same order,then λ_1 A±λ_2 B are also Skew-symmetric matrix
for λ_1 any λ_2 scalar .
Exp.(λ_1 A+λ_2 B)^T=λ_1 A^T+λ_2 B^T
=-λ_1 A-λ_2 B=-(λ_1 A+λ_2 B)
(vii) If A and B are both Skew-symmetric matrices of same order,then AB is Skew-symmetric iff AB-BA=0
Exp. (AB)^T=B^T A^T
=(-B) (-A)=BA=-AB
(viii) If A and B are symmetric matrices of same order,then AB-BA must be Skew-symmetric matrix .
(ix) If A is any Skew-symmetric matrix ,then all positive odd integral power of A are Skew-symmetric matrix.
(x) If A any square matric,then A-A^T is always Skew-symmetric.

Properties of Skew Symmetric Matrices linear algebra csir net gate iit jam mathematics msc entrance