Answer :

We are given that,


Where,


B = symmetric matrix


C = skew-symmetric matrix


We need to find B.


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


A symmetric matrix A = AT


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 AT = -A


So, let the matrix B 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 = (1 2)


2nd row of A = (0 3)


Transpose of this matrix A, AT will be given as


1st column of AT = 1st row of A = (1 2)


2nd column of AT = 2nd row of A = (0 3)


Then,



Substituting the matrix A and AT in B,







Taking transpose of B,


1st row of B = (1 1)


2nd row of B = (1 3)


Transpose of this matrix B, BT will be given as


1st column of BT = 1st row of B = (1 1)


2nd column of AT = 2nd row of A = (1 3)


Then,



Since, B = BT. Thus, B is symmetric.


Now, let the matrix C be



Substituting the matrix A and AT in C,







Multiplying -1 on both sides,





Taking transpose of C,


1st row of C = (0 1)


2nd row of C = (-1 0)


Transpose of this matrix C, CT will be given as


1st column of CT = 1st row of C = (0 1)


2nd column of CT = 2nd row of C = (-1 0)


Then,



Since, CT = -C. Thus, C is skew-symmetric.


Check:



Put the value of matrices B and C.





Matrices B and C satisfies the equation.


Hence, .


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