We are given that,

Where,

P = symmetric matrix

Q = skew-symmetric matrix

We need to find P.

A symmetric matrix is a square matrix that is equal to its transpose.

A symmetric matrix ⬄ P = P^T

Now, let us understand what skew-symmetric matrix is.

A skew-symmetric matrix is a square matrix whose transpose equals its negative, that, it satisfies the condition

A skew symmetric matrix ⬄ Q^T = -Q

So, let the matrix P be

Let us calculate A^T.

We know that the transpose of a matrix is a new matrix whose rows are the columns of the original.

We have,

Here,

1^st row of A = (3 5)

2^nd row of A = (7 9)

Transpose of this matrix A, A^T will be given as

1^st column of A^T = 1^st row of A = (3 5)

2^nd column of A^T = 2^nd row of A = (7 9)

Then,

Substituting the matrix A and A^T in P,

Taking transpose of P,

1^st row of P = (3 6)

2^nd row of P = (6 9)

Transpose of this matrix P, P^T will be given as

1^st column of P^T = 1^st row of P = (3 6)

2^nd column of P^T = 2^nd row of P = (6 9)

Then,

Since, P = P^T. Thus, P is symmetric.

Now, let the matrix Q be

Substituting the matrix A and A^T in Q,

Multiplying -1 on both sides,

Taking transpose of Q,

1^st row of Q = (0 -1)

2^nd row of Q = (1 0)

Transpose of this matrix Q, Q^T will be given as

1^st column of Q^T = 1^st row of Q = (0 -1)

2^nd column of Q^T = 2^nd row of Q = (1 0)

Then,

Since, Q^T = -Q. Thus, Q is skew-symmetric.

Check:

Put the value of matrices P and Q.

Matrices P and Q satisfies the equation.

Hence, .