Find the symmetric and skew-symmetric parts of the matrix A = [[1, 2, 4], [6, 8, 1], [3, 5, 7]]

To find the symmetric and skew-symmetric parts of the matrix A, we first need to understand what these terms mean. The symmetric part of a matrix is the part that remains unchanged when the matrix is transposed, while the skew-symmetric part changes sign when the matrix is transposed. For any square matrix A, we can express it as the sum of its symmetric and skew-symmetric parts.

Definitions

Let’s define the symmetric and skew-symmetric parts mathematically:

• S (symmetric part) = (A + A^T) / 2
• K (skew-symmetric part) = (A - A^T) / 2

Step 1: Calculate the Transpose of Matrix A

First, we need to find the transpose of matrix A. The transpose of a matrix is obtained by swapping its rows and columns. For the given matrix A:

A = [[1, 2, 4],
     [6, 8, 1],
     [3, 5, 7]]

The transpose A^T is:

A^T = [[1, 6, 3],
       [2, 8, 5],
       [4, 1, 7]]

Step 2: Calculate the Symmetric Part

Now, we can find the symmetric part S:

S = (A + A^T) / 2

Calculating A + A^T:

A + A^T = [[1+1, 2+6, 4+3],
            [6+2, 8+8, 1+5],
            [3+4, 5+1, 7+7]]
         = [[2, 8, 7],
            [8, 16, 6],
            [7, 6, 14]]

Now, dividing by 2:

S = [[2/2, 8/2, 7/2],
     [8/2, 16/2, 6/2],
     [7/2, 6/2, 14/2]]
   = [[1, 4, 3.5],
      [4, 8, 3],
      [3.5, 3, 7]]

Step 3: Calculate the Skew-Symmetric Part

Next, we find the skew-symmetric part K:

K = (A - A^T) / 2

Calculating A - A^T:

A - A^T = [[1-1, 2-6, 4-3],
            [6-2, 8-8, 1-5],
            [3-4, 5-1, 7-7]]
         = [[0, -4, 1],
            [4, 0, -4],
            [-1, 4, 0]]

Now, dividing by 2:

K = [[0/2, -4/2, 1/2],
     [4/2, 0/2, -4/2],
     [-1/2, 4/2, 0/2]]
   = [[0, -2, 0.5],
      [2, 0, -2],
      [-0.5, 2, 0]]

Final Results

In summary, the symmetric and skew-symmetric parts of the matrix A are:

• S (symmetric part):

          [[1, 4, 3.5],
           [4, 8, 3],
           [3.5, 3, 7]]
          

• K (skew-symmetric part):

          [[0, -2, 0.5],
           [2, 0, -2],
           [-0.5, 2, 0]]
          

This method of decomposition is quite useful in various applications, including physics and engineering, where understanding the properties of matrices is essential.