Symmetric Matrix & Skew Symmetric Matrix - Practice Questions & MCQ

Symmetric matrix:

A square matrix  is said to be symmetric if A' = A,

Clearly, A = A', hence A is a symmetric matrix

Skew-symmetric matrix:  

A square matrix  is said to be skew-symmetric if A’ = -A 

That means all the diagonal element of a skew-symmetric matrix are 0.

Properties of Symmetric and Skew-symmetric Matrices:

     i) If A is a square matrix, then AA’ and A’A are symmetric matrices

    ii) If A is a symmetric matrix, then -A, kA, A’, Aⁿ, B’AB are also symmetric matrix where n ∈ N, k ∈ R and B is a square matrix of order same as matrix A.

    iii) If A is a skew-symmetric matrix then   

1. A²ⁿ is a symmetric matrix for n ? N.

2. A²ⁿ⁺¹ is a skew-symmetric matrix for n ? N

3. kA is also a skew-symmetric matrix, where  k ∈ R

4. B’AB  is also a skew-symmetric matrix where B a square matrix of order same as matrix A

    iv) If A and B are symmetric matrices then:

1. A ± B, AB+BA are symmetric matrices.

2. AB - BA is a skew-symmetric matrix.

    v) If A and B are skew-symmetric matrices then:

1. A ± B, AB - BA are skew-symmetric matrices.

2. AB + BA is a symmetric matrix.