Question

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

Answer 1

Let A = [a_ij] be a matrix which is both symmetric and skew symmetric.

Since A is a skew symmetric matrix, so A′ = –A.

Thus for all i and j, we have a_ij = – a_ji. .........(1)

Again, since A is a symmetric matrix, so A′ = A.

Thus, for all i and j, we have

a_ji = a_ij .............(2)

Therefore, from (1) and (2), we get

a_ij = –a_ij for all i and j

or 2a_ij = 0,

i.e., a_ij = 0 for all i and j. Hence A is a zero matrix.