Question

A matrix which is both symmetric and skew symmetric matrix is a ______.

Options

• triangular matrix

• identity matrix

• diagonal matrix

• null matrix

Answer 1

A matrix which is both symmetric and skew symmetric matrix is a null matrix.

Explanation:

A matrix that is both symmetric and skew-symmetric must satisfy the properties of both types:

1. Symmetric Matrix: A matrix A is symmetric if A = A^T, meaning it is equal to its transpose.
2. Skew-Symmetric Matrix: A matrix A is skew-symmetric if A = – A^T, meaning it is equal to the negative of its transpose.

For a matrix to be both symmetric and skew-symmetric, we have:

A = A^T and A = – A^T

Combining these, we get:

A = – A

This implies that each element of the matrix must be zero:

A_ij = – A_ij

2A_ij = 0

A_ij = 0

Therefore, the only matrix that satisfies both conditions is the null matrix (a matrix where all elements are zero).