Linear Algebra, 2020a

260 Chapter Three. Maps Between Spaces
for example), but there are differences (failure of commutativity). This section
provides an example that algebra systems other than the usual real number one
can be interesting and useful.
Exercises
4.12 Supply the intermediate steps in Example 4.9.
̌ 4.13 Use
(
Corollary
)
4.11 to
(
decide
)
if each matrix
(
has
)
an inverse.
2 1 0 4
2 −3
(a)
(b)
(c)
−1 1 1 −3 −4 6
̌ 4.14 For each invertible matrix in the prior problem, use Corollary 4.11 to find its
inverse.
̌ 4.15 Find the inverse, if it exists, by using the Gauss-Jordan Method. Check the
answers for the 2×2 matrices with Corollary 4.11.
( ) ( ) ( )
3 1 2 1/2
2 −4
(a)
(b)
(c)
0 2 3 1
−1 2
⎛
⎞ ⎛
⎞
0 1 5
2 2 3
(e)
⎝
0 −2 4
2 3 −2
⎠
(f)
⎝
1 −2 −3
4 −2 −3
̌ 4.16 What matrix has this one for its inverse?
( ) 1 3
2 5
⎠
(d)
⎛
1 1
⎞
3
⎝ 0 2 4⎠
−1 1 0
4.17 How does the inverse operation interact with scalar multiplication and addition
of matrices?
(a) What is the inverse of rH?
(b) Is (H + G) −1 = H −1 + G −1 ?
̌ 4.18 Is (T k ) −1 =(T −1 ) k ?
4.19 Is H −1 invertible?
4.20 For each real number θ let t θ : R 2 → R 2 be represented with respect to the
standard bases by this matrix.
( )
cos θ − sin θ
sin θ cos θ
Show that t θ1 +θ 2
= t θ1 · t θ2 . Show also that t −1 θ = t −θ .
4.21 Do the calculations for the proof of Corollary 4.11.
4.22 Show that this matrix
( ) 1 0 1
H =
0 1 0
has infinitely many right inverses. Show also that it has no left inverse.
4.23 In the review of inverses example, starting this subsection, how many left
inverses has ι?
4.24 If a matrix has infinitely many right-inverses, can it have infinitely many
left-inverses? Must it have?