Linear Algebra, 2020a

Section II. Geometry of Determinants 361
(b) For the |S| ≠ 0 case, to show that |TS|/|S| = |T| for all transformations, consider
the function d: M n×n → R given by T ↦→ |TS|/|S|. Show that d has the first
property of a determinant.
(c) Show that d has the remaining three properties of a determinant function.
(d) Conclude that |TS| = |T| · |S|.
1.26 Give a non-identity matrix with the property that A T = A −1 . Show that if
A T = A −1 then |A| = ±1. Does the converse hold?
1.27 The algebraic property of determinants that factoring a scalar out of a single
row will multiply the determinant by that scalar shows that where H is 3×3, the
determinant of cH is c 3 times the determinant of H. Explain this geometrically,
that is, using Theorem 1.5. (The observation that increasing the linear size of a
three-dimensional object by a factor of c will increase its volume by a factor of c 3
while only increasing its surface area by an amount proportional to a factor of c 2
is the Square-cube law [Wikipedia, Square-cube Law].)
1.28 We say that matrices H and G are similar if there is a nonsingular matrix P
such that H = P −1 GP (we will study this relation in Chapter Five). Show that
similar matrices have the same determinant.
1.29 We usually represent vectors in R 2 with respect to the standard basis so vectors
in the first quadrant have both coordinates positive.
( )
⃗v
+3
Rep E2
(⃗v) =
+2
Moving counterclockwise around the origin, we cycle through four regions:
Using this basis
( ( ( (
+ − − +
··· −→ −→ −→ −→ −→ · · · .
+)
+)
−)
−)
( ( ) 0 −1
B = 〈 , 〉
1)
0
⃗β 2
⃗β 1
gives the same counterclockwise cycle. We say these two bases have the same
orientation.
(a) Why do they give the same cycle?
(b) What other configurations of unit vectors on the axes give the same cycle?
(c) Find the determinants of the matrices formed from those (ordered) bases.
(d) What other counterclockwise cycles are possible, and what are the associated
determinants?
(e) What happens in R 1 ?
(f) What happens in R 3 ?
A fascinating general-audience discussion of orientations is in [Gardner].
1.30 This question uses material from the optional Determinant Functions Exist
subsection. Prove Theorem 1.5 by using the permutation expansion formula for
the determinant.
̌ 1.31 (a) Show that this gives the equation of a line in R 2 through (x 2 ,y 2 ) and
(x 3 ,y 3 ).
x x 2 x 3
y y 2 y 3
∣1 1 1 ∣ = 0