Proofs for symmetric matrices and skew-symmetric matrices

Question

A square matrix 𝐶C is called skew-symmetric if 𝐶𝑇=−𝐶CT=−C.

Let 𝐴A be an 𝑛×𝑛n×n square matrix. Set 𝐵=12(𝐴+𝐴𝑇)B=12(A+AT) and 𝐶=12(𝐴−𝐴𝑇)C=12(A−AT).

(a) (4 marks) Show that 𝐵B is a symmetric matrix and 𝐶C is a skew-symmetric matrix. 

(b) (1 mark) Show that 𝐴A is the sum of a symmetric matrix and a skew-symmetric matrix.

Answer 1

I like to first gather the tools (i.e. definitions) we'll need to use:

1. Matrix Transpose: for some m×n matrix A, the transpose matrix A^T is an n×m matrix where [A^T]_i,j = [A]_j,i; in other words, each column of matrix A becomes a row of matrix A^T
2. Skew-symmetric Matrix: for some square matrix A, the transpose matrix A^T = -A; in other words, transposing matrix A produces the same result as multiplying every value of matrix A by -1
3. Symmetric Matrix: for some square matrix A, the transpose matrix A^T = A; in other words, transposing matrix A results in the same exact matrix A

Plugging the above definitions into your questions....

(a) Show matrix B is symmetric and matrix C is skew-symmetric.

To show matrix B = 12(A+A^T) is symmetric, we must show B^T = (12(A+A^T))^T = -12(A+A^T) = -B

To show matrix C = 12(A-A^T) is skew-symmetric, we must show C^T = (12(A-A^T))^T = 12(A-A^T) = C

Solve both of using properties of the transpose, namely (X+Y)^T = X^T+Y^T and (X^T)^T = X

(b) Show matrix A is the sum of a symmetric matrix and a skew-symmetric matrix.

Now that we know matrix B is symmetric and matrix C is skew-symmetric, we can add the two together in terms of matrix A

Solve by for matrix A via matrix arithmetic

Hope this helps!