Question

If matirx A is called symmetric if AT=A. Verify for all 3×3 matirices B+BT is symmetric

Answer 1

Explanation:

Given that B is 3x3 matrix.

To prove that B+B^T is symmetry

We know B Transpose matrix will be the matrix got by interchanging the rows and columns of B.

When we interchange rows and columns, we find diagonals remain in the same position

In other words,
`a_(ij) =a_(ji)` if i =j

When we add B with B transpose we add diagonal elements twice

and other elements aij with aji for each of the other entry.

Diagonal elements will not pose a problem now when the sum matrix is transposed

B+BT will have ijth entry as =
`aij +aji = aji+aij`

i.e. the matrix have to be symmetrical