Download 1. Find the area of the parallelogram whose vertices are P(1, 1, 1), Q

1. Find the area of the parallelogram whose vertices are P (1, 1, 1), Q = (2, 3, −3) and R = (2, 1, 5).
√
A. 8√ 2
B. 6√7
C. 2 33
D. 16
E. 18
F. 20
2. We consider a system of linear equations with augmented matrix A. Which combination of true/false is
correct?
• If the reduced row echelon form of A has a row of zeros, the system has infinitely many solutions.
• If the system has a solution, the rank of A and the rank of the coefficient matrix coincide.
• If there are more variables than equations, the system has infinitely many solutions.
A. true, true, false
B. true, false, true
C. false, true, true
D. true, false, false
E. false, true, false
F. false, false, true
Mini Test 1
MAT1341B (Sept. 30, 2004)
3. Find the (2, 3)-entry of the product
  3 4i
17i 2 − i
13
4
 5i 0
 0
2i
3 + 7i i  
17 i
2i
13
i
0
−3 5i


3+i
2i 
.
0
3i
A. 4-2i
B. 5+ 3i
C. 6i
D. -7
E. 8
F. -9
4. Only two of the following are true for 3 × 3-matrices. Which two?
(i) Matrix addition and matrix multiplication are associative.
(ii) Matrix addition is associative but matrix multiplication is not.
(iii) Matrix addition and matrix multiplication are commutative.
(iv) Matrix multiplication is commutative but matrix addition is not.
(v) If the matrix product AB = 0, it does not follow that A or B is a zero matrix.
A. (i) and (iii)
B. (iv) and (v)
C. (ii) and (iii)
D. (i) and (v)
E. (ii) and (iv)
F. (i) and (iv)
Page 2
Mini Test 1
MAT1341B (Sept. 30, 2004)
5. What is the distance from the point (-5, 0, 2) to the plane x − y = 5?
√
A. 5 √2
B. 10 2
C. 10
D. 2
E. 5√
F. 2 5
Page 3
Mini Test 1
MAT1341B (Sept. 30, 2004)
Page 4
6. (10 points) In the following linear system replace α by the last digit of your student number.
Let b and c be arbitrary real numbers and consider the following linear system
x
x
y
+ y
+ 2y
+
−
+
2b z
2α z
4b z
=
=
=
−2 − α
5−c
3−α
Find all values of b and c for which the system has
(i) a unique solution;
(ii) no solution;
(iii) infinitely many solutions.
In case (iii) determine what the solutions are. Record your answers below.
Marking: 2 points for row reduction containing variables; 2 point each for the correct answer to (i), (ii)
and (iii), 2 points for general solution.
Answers: (i) There is a unique solution exactly when
(ii) There is no solution exactly when
.
.
(iii) There are infinitely many solutions exactly when
and these solutions are [x, y, z]T =
.
.
Mini Test 1
MAT1341B (Sept. 30, 2004)
7. (5 points) Find the general solution and basic
whose coefficient matrix is

1
 −2
2
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solutions of the homogeneous system of linear equations

3 −2 −1
1
0
4 
13 −8 0
and express the general solution as a linear combination of your basic solutions.
Marking: 3 points for row reduction, 1 point for basic solutions and 1 point for linear combination.
Answers: The general solution is
basic solutions:
linear combination: