MAT1503
EXAM PACK
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Exam Preparation
ZAK I. Ali
Universityof SouthAfrica
Department
of Mathematics
MAT1503:LINEAR ALGEBRA I
Exercise 1
Question 1.
Assume we are given the system
x +y −z +2t = 2
2x +z −t = 3
−y +2t = 2.
Which of the following elements are solutions of the above system?
(1) (1,-1,0,1)
(2) (1,-1,0,1, 0)
(3) (0,1,3,1)
(4) (2,-3,3,1)
Question 2
Consider the system
x +2y =1
3x +ky =3
x +ky +z =2
(1) For which value(s) of k is (1, 0, 1) a solution to the above system?
(2) For which value(s) of k is (1, 0, −1) a solution to the above system?
(3) For which value(s) of k is (3, −1, 5) a solution to the above system?
Additional Exercises
Same instructions as Question 2.
1
Open Rubric
Downloaded by Jonah Njenga ()
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(1)
3 2 1
0k+1 3
0 0 k−3
(2)
λ 2 1
03 2
00λ+3
(3)
2 3 4
0k−1 3
0 0 (k − 1)(k + 2)
2
Downloaded by Jonah Njenga ()
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Exercise 2
Suppose A,B and C are 2 × 2 matrices such that det (A) = 3, −1) = −2 and
det(B
det(CT ) = 4.
Evaluate
(a) det (ABC)
T
(b) det [C2]
(c) det (−4A)
−1
(d) det (−4A )
−1
(e) det [−4A] .
Exercise 3
For which values ofk is the coefficientmatrix ofthe system given in Question 2,
Exercise 1 invertible?
Exercise 4
−1 5 −1 3
Let A = and B = ,
2 3 −2 1
(i) Compute
(a) A −1, B −1 and AB
(b) AA −1, BB −1 and A−1B −1
(c) A 2, B 2 and A2 − B2.
(ii) Using (a), determine a matrix Z such that ZA = B
(iii) Using (a) determine a matrix Y such that AY = B.
Exercise 5
(1) Solve the following system with Cramer’s Rule
2x1 + x 2 + x 3 = 3
x1 − x 2 − x 3 = 0
x1 + 2x 2 − x 3 = 0
(2) Use cofactors to find the determinant of the coefficient matrix of the system.
Exercise 6
~
b be the vectors given by ~a = h2, 0,~
Let ~a and b−1i
= h1,
and−1, 3i.
3
Downloaded by Jonah Njenga ()
EXAM PACK
, lOMoARcPSD|58918787
Exam Preparation
ZAK I. Ali
Universityof SouthAfrica
Department
of Mathematics
MAT1503:LINEAR ALGEBRA I
Exercise 1
Question 1.
Assume we are given the system
x +y −z +2t = 2
2x +z −t = 3
−y +2t = 2.
Which of the following elements are solutions of the above system?
(1) (1,-1,0,1)
(2) (1,-1,0,1, 0)
(3) (0,1,3,1)
(4) (2,-3,3,1)
Question 2
Consider the system
x +2y =1
3x +ky =3
x +ky +z =2
(1) For which value(s) of k is (1, 0, 1) a solution to the above system?
(2) For which value(s) of k is (1, 0, −1) a solution to the above system?
(3) For which value(s) of k is (3, −1, 5) a solution to the above system?
Additional Exercises
Same instructions as Question 2.
1
Open Rubric
Downloaded by Jonah Njenga ()
, lOMoARcPSD|58918787
(1)
3 2 1
0k+1 3
0 0 k−3
(2)
λ 2 1
03 2
00λ+3
(3)
2 3 4
0k−1 3
0 0 (k − 1)(k + 2)
2
Downloaded by Jonah Njenga ()
, lOMoARcPSD|58918787
Exercise 2
Suppose A,B and C are 2 × 2 matrices such that det (A) = 3, −1) = −2 and
det(B
det(CT ) = 4.
Evaluate
(a) det (ABC)
T
(b) det [C2]
(c) det (−4A)
−1
(d) det (−4A )
−1
(e) det [−4A] .
Exercise 3
For which values ofk is the coefficientmatrix ofthe system given in Question 2,
Exercise 1 invertible?
Exercise 4
−1 5 −1 3
Let A = and B = ,
2 3 −2 1
(i) Compute
(a) A −1, B −1 and AB
(b) AA −1, BB −1 and A−1B −1
(c) A 2, B 2 and A2 − B2.
(ii) Using (a), determine a matrix Z such that ZA = B
(iii) Using (a) determine a matrix Y such that AY = B.
Exercise 5
(1) Solve the following system with Cramer’s Rule
2x1 + x 2 + x 3 = 3
x1 − x 2 − x 3 = 0
x1 + 2x 2 − x 3 = 0
(2) Use cofactors to find the determinant of the coefficient matrix of the system.
Exercise 6
~
b be the vectors given by ~a = h2, 0,~
Let ~a and b−1i
= h1,
and−1, 3i.
3
Downloaded by Jonah Njenga ()
