AST11117 Introduction to Linear Algebra
Assignment 3
(To be submitted to Canvas no later than the end of 9 December 2017)
Question 1
Solve for a , b , c , and d in the following matrix equation
Answer
Question 2
Find x such that the matrix [ 2 𝑥−1 −2] equals to its own inverse.
Answer
Question 3
Find the inverse of matrix A , where A = [
3 2 −1
1 6 3
2 −4 0
] , by using elementary row operations.
Answer
Question 4
Find the inverse of matrix A , where A = [
3 2 −1
1 6 3
2 −4 0
] , by using the determinant and the
adjoint of A .
Answer
Question 5
Write the following system of linear equations in the form. Ax = b , and solve for x1 , x2 , x3 by
Gauss-Jordan Elimination.
x1 – x2 + 4x3 = 17
x1 + 3x2 = – 11
– 6x2 + 5x3 = 40
Answer
Question 6
Solve for x1 , x2 , x3 in the following system of linear equations by using the matrix of
cofactors and then the inverse of the coefficient matrix of the system
x1 + x2 – 2 x3 = 0
x1 – 2 x2 + x3 = 0
x1 – x2 – x3 = –1
Answer
Question 7
One 800-g glass of apple juice and one 800-g glass of orange juice contain a total of 177.4 mg of
vitamin C. Two 800-g glasses of apple juice and three 800-g glasses of orange juice contain a
total of 436.7 mg of vitamin C. How much vitamin C is in an 800-g glass of each type of juice?
Answer
Question 8
A man borrowed from a bank a total of $500,000 among which some at a simple annual interest
rate of 9%, some at a simple annual interest rate of 10%, and some at a simple annual interest
rate of 12%. The amount borrowed at 10% was 2.5 times the amount borrowed at 9%. Without
paying back any of the principals, he is paying a total interest of $52,000 per annum to the bank.
Determine how much he borrowed at each of the three different rates.
Answer
Question 9
A force of 10 kgf is pulling in the direction given by the orientation of v = .
Express the force in the component form. of a vector on the same coordinate system.
Answer
Question 10
A force of 3 kgf is pulling in the direction given by the orientation of v = 3i + 3j – k .
Express the force in the component form. of a vector on the same coordinate system.
Answer
Question 11
Three persons A, B, and C are pulling on a small size but heavy metallic sphere by ropes
attaching to the same point on the sphere. The pulling forces exerted by A and B are
FA = kgf and FB = kgf respectively, whilst the weight of
the sphere w = kgf. Determine the vector force Fc that C must pull in order to
keep the sphere stationary. Find also the magnitude of Fc . (Correct to integer value.)
Answer
Question 12
What are the geometric relationships between the three points namely
P = (2, 3, 1) , Q = (4, 2, 2) , and R = (8, 0, 4) in the 3-dimensional space.
Answer
Question 13
What are the geometric relationships between the three points namely
P = (2, 1, 0) , Q = (4, 1, 2) , and R = (4, 3, 0) in the 3-dimensional space.
Answer
Question 14
Given that a = , find a vector b such that Proj𝐛𝐚 = ⟨4,0⟩ .
Answer
Question 15
Find the area of the triangle with two of its sides formed by a = and b = .
Answer
Question 16
Find the area of the parallelogram with three of its vertices at
A = ( 1, 1, 0 ) , B = ( 0, – 2, 1 ) and C = ( 1, – 3, 0 ) .
Answer
Question 17
Find the line passing through the point P = ( 1, 2, 1 )
and parallel to the line L: x = 2 – 3t , y = 4 , z = 6 + t .
Answer
Question 18
Find the line passing through the point P = ( 2, 0, 1 )
and perpendicular to both vectors a = and b = .
Answer
Question 19
Find the line passing through the point P = ( 1, 2, – 1 )
and perpendicular to the plane 2x – y + 3z = 12 .
Answer
Question 20
Find an equation of the plane containing the points
P = ( 2, 0, 3 ) , Q = ( 1, 1, 0 ) and R = ( 3, 2, – 1 ) .
Answer
Question 21
Find an equation of the plane containing the point P = ( 0, – 2, – 1 )
and parallel to the plane – 2x + 4y = 3 .
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Question 22
Find the shortest distance between the point P = ( 2, 0, 1 ) and the plane 2x – y + 2z = 4 .
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Question 23
Find the line of intersection of the planes 3x + y – z = 2 and 2x – 3y + z = – 1 .
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Question 24
Find the shortest distance between the point P = ( 4, – 1, 5 ) and
the line L: x = 3 , y = 1 + 3t , z = 1 + t .
Answer
For Questions 25 – 29, 𝑖 = √−1 .
Question 25
Write 3+2𝑖2+5𝑖 in standard form. (i.e. in the form. of a + bi where a, b are real numbers).
Answer
Question 26
Solve for x where 𝑥2 + 𝑥 − 3 = (2𝑥 + 1)𝑖 .
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Question 27
Express (1 + √3𝑖)−4 in standard form.
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Question 28
Given that 𝑖 = √−1 , what are the square roots of i .
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Question 29
Find the fourth roots of √2 + √2𝑖 .
Answer