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 = 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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