If a_ij is symmetric and b_ij is antisymmetric, prove in general that the product a_ijb_ij is zero. Verify this result for the specific case by using the symmetric and antisymmetric terms from Exercise 1.2.

Data from exercise 1.2

Use the decomposition result (1.2.10) to express a_ij from Exercise 1.1 in terms of the sum of symmetric and antisymmetric matrices. Verify that a(_ij) and a[_ij] satisfy the conditions given in
the last paragraph.

Data from exercise 1.1

For the given matrix/vector pairs, compute the following quantities: a_ii, a_ija_ij, a_ija_jk, a_ijb_j, a_ijb_ib_j, b_ib_j, b_ib_i. For each case, point out whether the result is a scalar, vector or matrix. Note that aijbj is actually the matrix product [a]{b}, while a_ija_jk is the product [a][a].

Equation 1.2.10

aij (aij+aji) + (aij - aji) = a(ij) + a[ij]