A symmetric matrix satisfies P=PT (flipping across the main diagonal doesn't change it).
A skew-symmetric matrix satisfies P=−PT (flipping across the main diagonal gives the negative).
Since P is both symmetric and skew-symmetric:
P=PT (symmetric condition)
P=−PT (skew-symmetric condition)
From the symmetric condition, PT=P.
Substituting into the skew-symmetric condition:
P=−PT
P=−P
P=−P
P+P=0
2P=0
P=0
Therefore, P is a zero matrix (all elements are zero).
A diagonal matrix includes the zero matrix, but not all diagonal matrices are both symmetric and skew-symmetric.
A square matrix includes the zero matrix, but not all square matrices are both symmetric and skew-symmetric.
The identity matrix is symmetric but not skew-symmetric.
The zero matrix is the only matrix that can be both symmetric and skew-symmetric.
The answer is: P is a zero matrix.