# MA1101R-HW2 ```NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 2, 2016/2017
MA1101R Linear Algebra
Homework Assignment 2
Please write your name, student card number and tutorial group number on the
answer script, and submit during the lecture on 16th February 2017 (Thursday).
1. For each of the following matrices,
(i) Use Gauss-Jordan elimination to find its inverse.
(ii) Write down the elementary row operation used in each step, and the determinant
of the corresponding elementary matrix.
(iii) Find the determinant of the given matrix.
(a) A = (aij )3&times;3 , where aij =
1
;
i+j
(b) B = (bij )3&times;3 , where bij = |i + j| − |i − j|.


1 1 1 1


0 1 1 1

. Find adj(A) by using Definition 2.5.24.
2. Let A = 

0
0
1
1


0 0 0 1
3. Use Cramer’s rule to solve the following linear system:


 3x − 2y + 5z = 1
4x − y + z = 2


−2x + y + 2z = 3.
4. For each pair of S1 and S2 , determine whether (i) span(S1 ) ⊆ span(S2 ), and (ii)
span(S2 ) ⊆ span(S1 ).
(a) S1 = {(1, 0, 1, 0), (1, 2, 1, 2), (3, 1, 2, 1)}, S2 = {(1, 1, 1, 1), (2, 1, 2, 1), (3, 1, 3, 1)}.
(b) S1 = {(3, −2, 5, 1), (4, −1, 1, 2), (−2, 1, 2, 3)}, S2 = {(9, −4, 4, 0), (0, −2, 11, 3), (−1, −2, −1, 9)}.
5. Let A be a square matrix. Prove that there exists a nonzero square matrix B such
that AB = 0 if and only if det(A) = 0. [Hint: Use Theorems 2.4.7 and 2.5.19. Note
that matrices can be multiplied in blocks; see equation (2.2).]
6. Let A be a square matrix of order n that has exactly one nonzero entry in each row and
each column. Let D be the diagonal matrix whose ith diagonal entry is the nonzero
entry in the ith row of A.
1
MA1101R LINEAR ALGEBRA HOMEWORK ASSIGNMENT 2

0
0
a1

 a2
[For example, if A = 
0

0
0
0
0
0
a4
0
2


a1 0 0 0



 0 a2 0 0 
0

, then D = 
 0 0 a 0 .]
a3 
3



0 0 0 a4
0
0

Prove that
(i) A = DP for a permutation matrix P .
(ii) A−1 = AT D −2 .
[Hint: Refer to Homework Assignment 1, Problem 6. For (i), show that a permutation
matrix P can be obtained from A by a series of elementary row operations. For (ii),
use the result that P P T = I which was proved in Homework Assignment 1.]
```