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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 = PT

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 QT = -Q

So, let the matrix P be

Let us calculate AT.

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

We have,

Here,

1st row of A = (3 5)

2nd row of A = (7 9)

Transpose of this matrix A, AT will be given as

1st column of AT = 1st row of A = (3 5)

2nd column of AT = 2nd row of A = (7 9)

Then,

Substituting the matrix A and AT in P,

Taking transpose of P,

1st row of P = (3 6)

2nd row of P = (6 9)

Transpose of this matrix P, PT will be given as

1st column of PT = 1st row of P = (3 6)

2nd column of PT = 2nd row of P = (6 9)

Then,

Since, P = PT. Thus, P is symmetric.

Now, let the matrix Q be

Substituting the matrix A and AT in Q,

Multiplying -1 on both sides,

Taking transpose of Q,

1st row of Q = (0 -1)

2nd row of Q = (1 0)

Transpose of this matrix Q, QT will be given as

1st column of QT = 1st row of Q = (0 -1)

2nd column of QT = 2nd row of Q = (1 0)

Then,

Since, QT = -Q. Thus, Q is skew-symmetric.

Check:

Put the value of matrices P and Q.

Matrices P and Q satisfies the equation.

Hence, .

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