Exercises
In the following exercises if the size of the matrix is not specified, then assume
that the given matrices are square matrices.
10-1. Given the matrices: A=
[
3 7
1 2]
A−^1 =[
−2 7
1 − 3]B=[
5 8
3 5]
B−^1 =[
5 − 8
−3 5](a) Verify that AA −^1 =A−^1 A=I
(b) Verify that BB −^1 =B−^1 B=I
(c) Calculate AB
(d) Calculate B−^1 A−^1
(e) Find (AB )−^1 and check your answer.
10-2. Start with the definition AA −^1 =I and take the transpose of both sides of
this equation. Note that IT=Iand show that (A−^1 )T= (AT)−^1
10-3. Show that (ABC )−^1 =C−^1 B−^1 A−^1
Hint: ABC = (AB )C
10-4. If Aand Bare nonsingular matrices which commute, then show that
(a) A−^1 and Bcommute
(b) B−^1 and Acommute
(c) A−^1 and B−^1 commute Hint: If AB =BA, then A−^1 (AB )A−^1 =A−^1 (BA )A−^1
10-5. If Ais nonsingular and symmetric, show that A−^1 is also symmetric.
Hint: If AA −^1 =I, then (AA −^1 )T= (A−^1 )TAT=I
10-6. If Ais nonsingular and AB =AC , show B=C
10-7. Show that if AB =Aand BA =B, then Aand B are idempotent.
Hint: Examine the products ABA and BAB
10-8.