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 = A^{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 ⬄ A^{T} = -A

So, let the matrix B 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 = (1 2)

2^{nd} row of A = (0 3)

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

1^{st} column of A^{T} = 1^{st} row of A = (1 2)

2^{nd} column of A^{T} = 2^{nd} row of A = (0 3)

Then,

Substituting the matrix A and A^{T} in B,

Taking transpose of B,

1^{st} row of B = (1 1)

2^{nd} row of B = (1 3)

Transpose of this matrix B, B^{T} will be given as

1^{st} column of B^{T} = 1^{st} row of B = (1 1)

2^{nd} column of A^{T} = 2^{nd} row of A = (1 3)

Then,

Since, B = B^{T}. Thus, B is symmetric.

Now, let the matrix C be

Substituting the matrix A and A^{T} in C,

Multiplying -1 on both sides,

Taking transpose of C,

1^{st} row of C = (0 1)

2^{nd} row of C = (-1 0)

Transpose of this matrix C, C^{T} will be given as

1^{st} column of C^{T} = 1^{st} row of C = (0 1)

2^{nd} column of C^{T} = 2^{nd} row of C = (-1 0)

Then,

Since, C^{T} = -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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