Answer :

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, .

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