Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 161
(b) Use the prior item.
That similar matrices have the same determinant is immediate from the above two: det(P T P −1 ) =
det(P ) · det(T ) · det(P −1 ).
Four.I.1.14
One way is to count these areas
A
B
y 2
x 2 x 1
D
y 1
C
E
F
by taking the area of the entire rectangle and subtracting the area of A the upper-left rectangle, B
the upper-middle triangle, D the upper-right triangle, C the lower-left triangle, E the lower-middle
triangle, and F the lower-right rectangle (x 1 +x 2 )(y 1 +y 2 )−x 2 y 1 −(1/2)x 1 y 1 −(1/2)x 2 y 2 −(1/2)x 2 y 2 −
(1/2)x 1 y 1 − x 2 y 1 . Simplification gives the determinant formula.
This determinant is the negative of the one above; the formula distinguishes whether the second
column is counterclockwise from the first.
Four.I.1.15 The computation for 2×2 matrices, using the formula quoted in the preamble, is easy. It
does also hold for 3×3 matrices; the computation is routine.
Four.I.1.16
No. Recall that constants come out one row at a time.
( ) ( )
( )
2 4
1 2
1 2
det( ) = 2 · det( ) = 2 · 2 · det( )
2 6
2 6
1 3
This contradicts linearity (here we didn’t need S, i.e., we can take S to be the zero matrix).
Four.I.1.17
Bring out the c’s one row at a time.
Four.I.1.18 There are no real numbers θ that make the matrix singular because the determinant of the
matrix cos 2 θ + sin 2 θ is never 0, it equals 1 for all θ. Geometrically, with respect to the standard basis,
this matrix represents a rotation of the plane through an angle of θ. Each such map is one-to-one —
for one thing, it is invertible.
Four.I.1.19 This is how the answer was given in the cited source. Let P be the sum of the three
positive terms of the determinant and −N the sum of the three negative terms. The maximum value
of P is
9 · 8 · 7 + 6 · 5 · 4 + 3 · 2 · 1 = 630.
The minimum value of N consistent with P is
9 · 6 · 1 + 8 · 5 · 2 + 7 · 4 · 3 = 218.
Any change in P would result in lowering that sum by more than 4. Therefore 412 the maximum value
for the determinant and one form for the determinant is
9 4 2
3 8 6
∣5 1 7∣ .
Subsection Four.I.2: Properties of Determinants
∣ ∣ 3 1 2
∣∣∣∣∣ 3 1 2
∣∣∣∣∣
Four.I.2.7 (a)
3 1 0
∣0 1 4∣ = 3 1 2
0 0 −2
0 1 4 ∣ = − 0 1 4
0 0 −2∣ = 6
1 0 0 1
1 0 0 1
1 0 0 1
(b)
2 1 1 0
−1 0 1 0
=
0 1 1 −2
0 0 1 1
=
0 1 1 −2
0 0 1 1
= 1
∣ 1 1 1 0∣
∣0 1 1 −1∣
∣0 0 0 1 ∣