Page 157 - 35Linear Algebra

7.6 Review Problems 157

(a) Compute:
T
i. AA ,
ii. AA T −1 ,

iii. B := A T AA T −1
(b) Show that the matrix B above is a right inverse for A, i.e., verify
that
AB = I .

(c) Is BA defined? (Why or why not?)
(d) Let A be an n × m matrix with n > m. Suggest a formula for a
left inverse C such that
CA = I

T
Hint: you may assume that A A has an inverse.
(e) Test your proposal for a left inverse for the simple example

1
A = ,
2

(f) True or false: Left and right inverses are unique. If false give a
counterexample.

Hint

5. Show that if the range (remember that the range of a function is the
set of all its outputs, not the codomain) of a 3 × 3 matrix M (viewed
3
3
as a function R → R ) is a plane then one of the columns is a sum of
multiples of the other columns. Show that this relationship is preserved
under EROs. Show, further, that the solutions to Mx = 0 describe this
relationship between the columns.

6. If M and N are square matrices of the same size such that M −1 exists
and N −1 does not exist, does (MN) −1 exist?

7. If M is a square matrix which is not invertible, is e M invertible?

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