MATH140 PennState University Linear Algebra Homework 7

Label the following statements as True or False. If the statementis False, give a counterexample.(a) If two rows of a matrix A are identical, then det(A) = 0.(b) If B is a matrix obtained from a square matrix A by multiplying a row of A by ascalar, then det(B) = det(A).(c) If B is a matrix obtained from a square matrix A by interchanging any two rows,then det(B) = −det(A).(d) If B is a matrix obtained from a square matrix A by adding k times row i to rowj, then det(B) = kdet(A).(e) If A ∈ Mn×n(R) has rank n, then det(A) = 0. 3. (10 points/5 each) Use the permutation formula from the end of lecture 15 to computethe determinants of the following matrices. Then, use the pivot formula to verify yourresult. Clearly show all solution steps. Linear Algebra (MATH140/MA-UY3044)
Fall 2019
Homework 7: sections 4.4, 5.1, 5.2 (lectures 14, 15)
(Due Wednesday 11/6/2019 before midnight)

1. (10 points) Consider the 2 × 2 matrix A =


a22 −a12
the matrix C =
. Prove that
−a21
a11

a11 a12
. The classical adjoint of A is
a21 a22
(a) (4 points) CA = AC = (detA)I, where I is the 2 × 2 identity matrix.
(b) (1 point) det(C) = det(A).
(c) (2 points) The classical adjoint of AT is C T .
(d) (3 points) If A is invertible, then A−1 =
1
C.
detA
2. (10 points/2 each) Label the following statements as True or False. If the statement
is False, give a counterexample.
(a) If two rows of a matrix A are identical, then det(A) = 0.
(b) If B is a matrix obtained from a square matrix A by multiplying a row of A by a
scalar, then det(B) = det(A).
(c) If B is a matrix obtained from a square matrix A by interchanging any two rows,
then det(B) = −det(A).
(d) If B is a matrix obtained from a square matrix A by adding k times row i to row
j, then det(B) = kdet(A).
(e) If A ∈ Mn×n (R) has rank n, then det(A) = 0.

3. (10 points/5 each) Use the permutation formula from the end of lecture 15 to compute
the determinants of the following matrices. Then, use the pivot formula to verify your
result. Clearly show all solution steps.





1 −2
3
−1
3 2
2 −5 
(a) A =  −1
(b) B =  4 −8 1 
3 −1
2
2
2 5
Linear Algebra (MATH140/MA-UY3044)
Fall 2019
4. (10 points/5 each) Apply the Gram-Schmidt process to the given subset S to obtain
an orthogonal basis β for span S. Then normalize the vectors in this basis to obtain
an orthonormal basis β̃ for span S.
(a) S = {(1, 0, 1), (0, 1, 1), (1, 3, 3)}
(b) S = {(2, −1, −2, 4), (−2, 1, −5, 5), (−1, 3, 7, 11)}
5. (10 points) Find the QR decomposition of the following matrices.


5
9
 1
7 

(a) (4 points) A = 
 −3 −5 .
1
5



(b) (6 points) B = 



1
2
5
−1
1 −4 

−1
4 −3 
.
1 −4
7 
1
2
1