National University of Singapore
Department of Mathematics
Semester 1, 2023/24
MA2001 Linear Algebra I
Tutorial 1
Questions 1-5 of this tutorial sheet will be discussed in the tutorial classes in Week 3
(28/8-1/9).
You are advised to revise Chapter 1 before attempting the questions.
Questions
1. Write down a general solution for each of the following linear equations.
(a) 4x + y = 1 in 2 variables x and y.
(b) x1 + x2 − 2x4 = 0 in 4 variables x1 , x2 , x3 and x4 .
(c) 2v − w + 2x − 4y + 6z = 8 in 5 variables v, w, x, y and z.
2. Each equation in the following linear system represents a line in the xy-plane:

 a1 x + b 1 y = c 1
a x + b2 y = c 2
 2
a3 x + b 3 y = c 3 ,
where a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 are constants and for each i = 1, 2, 3, ai , bi
are not both zero. Discuss the relative positions of the three lines when the system
(a) has no solution;
(b) has only one solution;
(c) has infinitely many solutions.
For each of the cases above, determine the reduced row-echelon form of the augmented matrix of the linear system.
3. (a) Does an inconsistent linear system with more unknowns than equations exist?
(b) Does a linear system which has only one solution, but more equations than
unknowns, exist?
(c) Does a linear system which has only one solution, but more unknowns than
equations, exist?
(d) Does a linear system which has infinitely many solutions, but more equations
than unknowns, exist?
Justify your answers.
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4. For each of the following augmented matrices, (i) determine whether the matrix
is in row-echelon form, reduced row-echelon form, both, or neither; (ii) find a
system of linear equations corresponding to the augmented matrix and (iii) solve
the system (if possible). You may assume that the variables are x1 , x2 , x3 , etc.




1 0 0 5
1 0 3 0
(a)  0 0 1 3  ,
(b)  0 −1 2 0  ,
1 1 0 4
0 0 1 0




1 0 0 0
−2 0 −1 −7 8
(c)  0 1 −1 0  ,
(d)  0 3 0
3
2 ,
0 0 0 1
0 0 0
1 −1




1 0 2 −2 3 −2
1 0 −2 0 2 0 −2




 0 0 1 1 3 2 
 0 1 0 0 2 0 4 
(e) 
 , (f) 
.
 0 0 0 0 0 0 
 0 0 0 1 −1 0 1 
0 0 0 1 5 5
0 0 0 0 0 1 1
5. The following is a system of linear equations in variables w, x, y and z :


w+ x+ y+ z = 1



aw + bx
=0

w + x + 2y + 2z = 3



ay + bz = 0
where a and b are constants.
(a) Write down the augmented matrix of the linear system and use Gaussian
Elimination to reduce the augmented matrix to a row-echelon form.
(Hint: You may need to consider the following three cases seperately:
(i) a 6= b; (ii) a = b 6= 0; and (iii) a = b = 0.)
(b) Determine the values of a and b so that the linear system is inconsistent.
(c) If a = b = 0, solve the linear system and write down a general solution.
More Exercises: (These questions may not be discussed in the tutorial class.)
6. In a three-commodity market, the supply and demand for each commodity depend
on the prices of the commodities. Let D1 , D2 , D3 be the demands for products 1,
2 and 3 respectively, S1 , S2 , S3 the respective supply and P1 , P2 , P3 the respective
prices for the commodities. Suppose the market can be described by the linear
equations
D1 = −2P1 + P2 + P3 + 4
D2 =
P1 + 2P2 + P3 − 1
D3 = 2P1 + P2 + P3 + 4
2
S1 = −P1 + 2P2 + 2P3 − 1
S2 = 2P1 + P2 + 2P3 − 2
S3 = P1 + 2P2 + 3P3 − 1.
Use Gaussian Elimination or Gauss-Jordan Elimination to find the equilibrium
solution (where supplies equal to demands).
7. Consider the homogeneous system of equations
ax + by + cz = 0
dx + ey + f z = 0
where a, b, c, d, e, f are constants.
(a) Let x = x0 , y = y0 , z = z0 be a solution to the system and k is a constant.
Show that x = kx0 , y = ky0 , z = kz0 is also a solution to the system.
(b) Let x = x0 , y = y0 , z = z0 and x = x1 , y = y1 , z = z1 be two solutions
to the system. Show that x = x0 + x1 , y = y0 + y1 , z = z0 + z1 is also a
solution to the system.
(c) Explain why if the system has a nontrivial solution, then the system has
infinitely many solutions.
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