Which of the following statements is/are true

1. Null matrix of order n is both symmetric and skew-symmetric matrix.

2. Null matrix of order n is both hermitian and skew hermitian matrix.

Concept: 

Symmetric Matrix:

Any real square matrix A = (a_ij) is said to be a symmetric matrix if and only if a_ij = a_ji, ∀ i, and j or in other words, we can say that if A is a real square matrix such that A = A’ then A is said to be a symmetric matrix.

Skew-symmetric Matrix:

Any real square matrix A = (a_ij) is said to be a skew-symmetric matrix if and only if a_ij = - a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A =- A’ then A is said to be a skew-symmetric matrix.

Hermitian Matrix:

Any square matrix says A is said to be a hermitian matrix if A = Aθ where Aθ is the transpose of the conjugate matrix of A.

Note: All the diagonal elements of a hermitian matrix are purely real.

Skew Hermitian Matrix:

Any square matrix say A is said to be a skew hermitian matrix if A = - Aθ where Aθ is the transpose of the conjugate matrix of A.

Note: All the diagonal elements of a skew hermitian matrix is purely imaginary or zero.

Calculation:

Statement 1: Null matrix of order n is both a symmetric and skew-symmetric matrix.

By the definition of a symmetric and skew-symmetric matrix, we can say that a null matrix of order n (As all the entries are 0) is both a symmetric and skew-symmetric matrix.

Hence, statement 1 is true.

Statement 2: Null matrix of order n is both a hermitian and skewed hermitian matrix.

By the definition of the hermitian and skew hermitian matrix, we can say that a null matrix (As all the entries are 0) of order n is both a symmetric and skew-symmetric matrix.

Hence, statement 2 is also true.

Additional Information

Properties of symmetric and Skew - Symmetric Matrices

1. In a skew-symmetric matrix A, all diagonal elements are zero. In fact, if A is skew-symmetric, then
   • a_ij = - a_ji for all i and j.
   • ⇒ a_ii = - a_ii
   • ⇒ a_ii = 0.
2. The matrix which is both symmetric and skew-symmetric must be a null matrix. In fact, if A = [a_ij] is symmetric, then
   • aij = aji for all i and j.
   • Further, if A = [aij] is skew-symmetric, then aij = - a_ji for all i and j. Adding, we get 2a_ij = 0 for all i and j so a_ij = 0 for all i and j.
   • Hence, A is a null matrix.
   • Thus, the "Null matrix is the only matrix which is both symmetric and skew-symmetric."