MACS-332A - Linear Algebra
Homework #6: 3.1-3.2, 4.1
True or False. You must justify your answer.
1. The cofactor expansion of detA down a column is the negative of the cofactor expansion along a row.
False: See Theorem 3.1
2. The determinant of a triangular matrix is the sum of the entries on the main diagonal.
False: See Theorem 3.2
3. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (−1)r , where r
is the number of row interchanges made during row reduction from A to U .
True: See discussion following example 2 on page 193.
4. If the columns of A are linearly dependent, then detA = 0.
True: See discussion following Theorem 3.4 on page 194.
5. A vector space is also a subspace.
True: See discussion preceding example 6 on page 220.
6. R2 is a subspace of R3 .
False: See example 8 on page 220.
1. Using the determinants of the following elementary matrices,





1 0 0
1 0 0
E1 =  0 1 0 , E2 =  0 1 0 , E3 = 
0 k 1
k 0 1





0 1 0
1 0 0
E4 =  0 k 0 , E5 =  1 0 0 , E6 = 
0 0 1
0 0 1
k
0
0
0
0
1
0
1
0
0
1
0

0
0 ,
1

2
0 
0
Answer the following questions using E1 , . . . , E6 to justify your responses.
(a) What is the determinant of an elementary row interchange matrix?
The determinant of an elementary row interchange matrix is −1 as shown by E5 ⇒ det(E5 ) = −1.
(b) What is the determinant of an elementary scaling matrix with k on the diagonal?
The determinant is k as shown by matrices, E3 and E4 .
(c) What is the determinant of an elementary row replacement matrix?
The determinant of an elementary row replacement matrix is 1 as shown by matrices, E2 and E1 .


−1 2 3 0
 3 4 3 0 

2. Using an appropriate cofactor expansion, find the detA where A = 
 5 4 6 6 
4 2 4 3
Cofactor expansion using
column
4:
−1 2 3 −1 2 3 det(A) = 0 + 0 − 6 · 3 4 3 + 3 · 3 4 3 4 2 4 5 4 6 = −6[−1(16 − 6) − 2(12 − 12) + 3(6 − 16)] + 3[−1(24 − 12) − 2(18 − 15) + 3(12 − 20)]
= −6[−10 − 30] + 3[−12 − 6 − 24] = 240 − 126
= 114
1
3. The following questions illustrate some important properties of the determinant.
1
det(A)
A · A−1 = I ⇒ det(AA−1 ) = det(I) = 1 We know that det(AA−1 ) = det(A) · det(A−1 )
1
⇒ det(A) · det(A−1 ) = 1 ⇒ det(A−1 ) =
det(A)
(a) Show that if A is invertible, then det A−1 =
(b) Let A and P be square matrices such that P −1 exists. Show that det(P AP −1 ) = det(A).
det(P )det(A)
det(P AP −1 ) = det(P )det(A)det(P −1 ) =
= det(A)
det(P )
(c) Let U be a square matrix such that U T U = I. Show that det(U ) = ±1.
Since U T U = I, det(U T U ) = det(I) = 1.
Since det(U T U ) = det(U T )det(U ) = det(U )det(U ) = [det(U )]2 = 1 ⇒ det(U ) = ±1.


2t
4. Let H be the set of all vectors of the form  0 . Find a vector v ∈ R3 such that H = Span{v}. Explain
−t
3
why
this
is
sufficient
to
show
that
H
is
a
subspace
of R

 . 



2
2
2t
 0  = t  0 . Thus, H = Span{v} where v =  0 .
−1
−1
−t


2
Since v =  0  ∈ R3 , we know that the Span{v} is a subspace of R3 .
−1
2