MATHEMATICAL METHODS FOR ENGINEERS 2 and LINEAR ALGEBRA
Assignment 1 – due Thurs 22nd August, 4pm
Please use a cover sheet and submit (together with the assignment questions from tutorial 2)
via the MME2 (or Linear Algebra, if appropriate) assignment box in the OC building. Make
sure you include your tutor’s name and tutorial day/time on your cover sheet.
Late submission incurs a penalty of 10% per day, up to 40%. Work more than 4 days late will
not be accepted.
1. (a) For the following matrices, calculate their determinant and find the inverse if it
exists.
(i) A =
!
1 1
2 1
"
(ii) B =
#
$
%
1 !2 3
!2 4 !6
1 0 !1
&
'
(
(b) Consider the following matrix:
A =
#
$
%
1 1 2
2k 1 4k
!1 0 !k
&
'
(
i. For what values of k does the inverse A!1 exist?
ii. Find A!1 for the case when k = 1.
Tutorial 1. Week 2
1. Given vectors !"a = (1 , !1 , 3),
!"
b = (2 , 1 , 0) and !"c = (!2 , 0 , !1). Find the dot product
of !"a and
!"
b , then !"c and !"a .
2. Find AB, BA, AT , A + BT if
A =
)
1 2 !1
0 !1 !3
*
and B =
+
,
-
1 0
!1 2
!3 4
.
/
0
.
3. Suppose that A, B, C, D and E are matrices of the following sizes:
Am"n, Bm"n, Cn"p, Dm"p, En"m.
Determine which of the following matrix expressions are defined. For those that are
defined, give the size of the resulting matrix F.
(a) F = BAT (b) F = AC + D (c) F = AE + B (d) F = AB + B
(e) F = E(A + B) (f) F = E(AC) (g) F = ET A (h) F = (AT + E)D
4. Given
A =
+
,
-
1 0 !1
2 !1 !3
!3 4 1
.
/
0
, E1 =
+
,
-
0 1 0
1 0 0
0 0 1
.
/
0
, E2 =
+
,
-
1 0 0
0 1 1
0 0 1
.
/
0
Find E1A and E2A. How do these multiplications change matrix A?
MATHEMATICAL METHODS FOR ENGINEERS 2 and LINEAR ALGEBRA
Assignment 1 – due Thurs 22nd August, 4pm
Please use a cover sheet and submit (together with the assignment questions from tutorial 1)
via the MME2 (or Linear Algebra, if appropriate) assignment box in the OC building. Make
sure you include your tutor’s name and tutorial day/time on your cover sheet.
Late submission incurs a penalty of 10% per day, up to 40%. Work more than 4 days late will
not be accepted.
2. Consider the following system of 3 equations:
x + y + 2z = 2
2x + y + 2z = 4
!x ! z = !1
(a) Write the system in the form Ax = b.
(b) Calculate det(A) = |A|. Is there a solution to the system of equations? Why or why
not?
(c) By perfoming row operations on the augmented matrix, find the solution the above
system of equations (if it exists).
3. This question will be similar to either Q2 or Q4 from Tutorial 2. It will be given to you
during your tutorial in week 4. It should be completed and submitted to your tutor at the
end of the allocated time. The relevant section of the formula sheet on the course website
will be provided – no other notes are allowed, no calculators allowed. Note that
each tute class will have a slightly di!erent question.
Tutorial 2.
1. In R3 the equation of the plane is
ax + by + cz = d.
Given three points, (2, !2, !3), (5, 2, 1), and (!1, 5, 4), find the equation of the plane
that passes through these three points.
2. Solve the following systems of linear, nonhomogeneous equations:
(a)
x + 2y + 3z = 6
4x + 5y + 6z = 15
7x + 8y + 9z = 24
(b)
w ! x + y ! z = 3
w + x + y + z = 5
w + 2x + 4y + 8z = 15
w + 3x + 9y + 27z = 41
Then, for each of the above four questions, solve the corresponding homogeneous linear
system.
Additional questions:
Kleinfeld and Kleinfeld, Section 1.2.1, pp. 18, 19, Questions 2, 6. To avoid having to do
the row operations, you can assume the RREF of the augmented matrices that are given
in the solutions in the back of the book.
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