Linear Algebra, 2020a

Section II. Geometry of Determinants 359
t( ⃗w)
∣ t(⃗v) ∣∣∣∣
3 3
−4 −2∣ = 6
1.7 Corollary If a matrix is invertible then the determinant of its inverse is the
inverse of its determinant |T −1 | = 1/|T|.
Proof 1 = |I| = |TT −1 | = |T| · |T −1 |
QED
Exercises
1.8 Is
⎛ ⎞
4
⎝1⎠
2
inside of the box formed by these three?
⎛ ⎞ ⎛ ⎞
3 2
⎝3⎠
⎝6⎠
1 1
⎛ ⎞
1
⎝0⎠
5
̌ 1.9 Find
(
the
(
volume
)
of the region defined by the vectors.
1 −1
(a) 〈 , 〉
3)
4
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
2 3 8
(b) 〈 ⎝1⎠ , ⎝−2⎠ , ⎝−3⎠〉
0 4 8
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 2 −1 0
(c) 〈 ⎜2
⎟
⎝0⎠ , ⎜2
⎟
⎝2⎠ , ⎜ 3
⎟
⎝ 0 ⎠ , ⎜1
⎟
⎝0⎠ 〉
1 2 5 7
̌ 1.10 In this picture the rectangle on the left is defined by the points (3, 1) and (1, 2).
Apply the matrix to get the rectangle on the right, defined by (7, 1) and (4, 2).
Why doesn’t this contradict Theorem 1.5?
( )
21
01
−→
area is 2 determinant is 2 area is 3
̌ 1.11 Find the volume of this region.
̌ 1.12 Suppose that |A| = 3. By what factor do these change volumes?