LINEAR ALGEBRA (Math 211)

**HW #7** (Due Wed. Oct. 18 )

**1.** For matrix
compute
.

**2.** Let *A* and *B* be 3 x 3 matrices with
and
.
Find the value of

**3.** Let *E*_{1}, *E*_{2}, *E*_{3} be 3 x 3 elementary matrices of
types (1), (2), and (3), respectively, and let *A* be a 3 x 3
matrices with .
Assume, additionally, that *E*_{1} corresponds
to multiplication of the second row by 3. Find the values of each of the
following

**4.** Assuming that
compute the following determinants

**5.**

(**a**) A matrix *A* is said to be *idempotent* if *A*^{2}=*A*.
Prove
that if *A* is idempotent, then
or 1.

(**b**) A matrix *A* is said to be *nilpotent* if *A*^{k}=0 for
some positive integer *k*. Prove
that if *A* is nilpotent, then .

(**c**) A matrix *A* is said to be *orthogonal* if
.
Prove that if *A* is orthogonal, then
or -1.

(**d**) An *n* x *n* matrix *A* is said to be *skew
symmetric*
if *A*^{T}=-*A*. Prove that if *A* is sew symmetric and *n* is odd, then
.