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Solution to Problem16


*****

Problem16:

A square matrix A is called skew-symmetric if,

> ;

                                     T
                                    A  = -A

Show that
  1. If A is an invertible skew-symmetric matrix, then its inverse is also skew-symmetric.
  2. If A and B are skew-symmetric, then so are A^T, A+B, A-B and kA for any scalar k.
  3. Every square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

SOLUTION:


Let's show that the inverse of A is skew-symmetric when A is. Since the operation of taking the transpose commutes with the operation of taking the inverse we have,

> ;

                                  -1 T     T -1
                                (A  )  = (A )

since A is skew-symmetric we can write,

> ;

                                  -1 T       -1
                                (A  )  = (-A)

and now using the obvious fact that for every non-zero scalar t,

inverse(t A) = (1/t) inverse(A)

we have,

> ;

                                   -1 T     -1
                                 (A  )  = -A

Thus, the inverse of A is skew-symmetric.

We now show that when A and B are skew-symm. so are the transpose of A, A+B, A-B and kA. We prove this by showing that the transpose of each of this matrices is equal to minus the matrix.

> ;

                                   T T       T      T
                                 (A )  = (-A)  = - A

this is due to the fact that A is skew-symmetric and the transpose of a number times a matrix is the number times the transpose of the matrix.

Now for A+B,

> ;

                         T    T    T
                  (A + B)  = A  + B  = -A - B = -(A + B)

idem for A-B,

> ;

                         T    T    T
                  (A - B)  = A  - B  = -A + B = -(A - B)

finally for kA, with k scalar,

> ;

                      T       T
                  (kA)  = k (A ) = k (-A) = - (kA)

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Feb 9 17:02:30 EST 2000