Homework3(due Wednesday 25th Oct midnight)

Q1. Write a system of equations that is equivalent to the given vector equations. a.

b.

Q2. Write a vector equation that is equivalent to the given system of equations.

Q3. Check if the following set of vectors form an orthogonal set in R3.

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ICS 6N – Homework 7

(In order to solve these problems, it is highly recommended to read Chapters 1.3 -1.5 in the book)

1. Write a system of equations that is equivalent to the given vector equation.

a)

b)

2. Write a vector equation that is equivalent to the given system of equations.

a)

b)

3. Check if the column vectors of matrix A are linearly independent or not.

a)

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7. Calculate the angle between the vectors u and v.

a. 𝒖 = [304

] , 𝒗 = [30

−4]

b. 𝒖 = [101

] , 𝒗 = [040

]

c. 𝒖 = [111

] , 𝒗 = [222

]

8. Calculate the projection vector c of vector b onto vector a.

a. 𝒂 = [100

] , 𝒃 = [321

]

b. 𝒂 = [030

] , 𝒃 = [12

−1]

c. 𝒂 = [002

] , 𝒃 = [503

]

9. Check if the following set of vectors form an orthogonal set in R3.

a. {[00

−2] , [

030

] , [−700

] , [102

] }

b. {[11

−2] , [

534

] , [402

] }

c. {[101

] , [20

−2] , [

030

] }

10. Verify properties of scalar product presented in Slide 3 of Lecture 3 for the following vectors and scalars:

𝒖 = [−121

] , 𝒗 = [122

] , 𝒘 = [111

] , 𝑐 = 2

Q4. Use determinant to find our if the matrix is invertible:

Q5. Compute the inverse of the following matrices if it exists:

Q6. Choose the values of h and k such that the system has: (i) no solution, (ii) unique solution,

(iii) many solutions. Given separate answers for each part.

(In order to solve problems 6 - 8, it is highly recommended to read Chapters 3.1, 3.2)

6. Find the determinants of the following matrices by using row reduction to echelon form:

a)

b)

c)

7.

a) Use determinant to find out if matrix is invertible:

b) Use determinant to find out if set of vectors is linearly independent:

ICS 6N – Homework 5

1. Compute the determinant of the following matrices:

a) 𝐴 = [ 3 −6−1 2 ]

b) 𝐵 = [5 21 3]

c) 𝐶 = [2 1 31 −1 −2

−1 3 2]

d) 𝐷 = [1 0 00 2 00 0 −4

]

e) 𝐸 = [1 2 4 32 4 8 6

−3 5 2 2−1 2 5 −4

]

2. Compute the inverse of the following matrices if it exists:

a) 𝐴 = [5 64 5]

b) 𝐵 = [6 77 8]

c) 𝐶 = [−5 10−4 8 ]

d) 𝐷 = [0 1 00 0 −1

−1 0 0]

e) 𝐸 = [−2 1 2 1−4 2 4 22 1 −3 3

−1 5 4 2]

10. Choose values of h and k such that the system has: I) no solution , II) unique solution, III) many solutions. Give separate answers for each part:

a)

b)

11. Mark each statement with True or False. Justify your answer.

a) In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of operations.

b) The row reduction algorithm applies only to augmented matrices for a linear system.

c) A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

d) Finding a parametric description of the solution set of a linear system is the same as solving the system.

e) If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.

f) The reduced row echelon form of a matrix is unique.

g) If every column of an augmented matrix contains a pivot, then the corresponding system is consistent.

h) The pivot positions in the matrix depend on whether row interchanges are used in the row reduction process.

i) A general solution of a system is an explicit description of all solutions of the system.

j) Whenever a system has free variables, the solution set contains many solutions.

Q7. Let A and B be 3x3 matrices with determinant of A = 4 and determinant of B = -3. Use

properties of determinant to compute:

a det (AB)

b det (5A)

c det BT

d det A-1

e det A3

Q8. Let A be a 4x4 matrix with determinant 7. Give a proof or a counter example for each of

the following.

Q9. Given three points A = (1, 3, 1), B = (2, 5, -3), C = (-4, 1, 8).

a. Find the angle in degrees at vertex A.

b. Find the area within the triangle formed by the three points.

Q10. Consider the two vectors v = 3i - 2j + 4k and w = i + j + 2k.

a. Find a vector u that is perpendicular to both v and w.

b. Find the area of the parallelogram that has adjacent edges given by v and w.

c. Find the area of the triangle that has adjacent edges given by v and w.

17. Let A : Rn → Rk be a linear map. Show that the following are equivalent.

a) A is injective (hence n ≤ k ). [injective means one-to-one]

b) dim ker(A) = 0.

c) A has a left inverse B , so BA = I .

d) The columns of A are linearly independent.

18. Let A : Rn → Rk be a linear map. Show that the following are equivalent.

a) A is surjective (hence n ≥ k ).

b) dim im(A) = k .

c) A has a right inverse B , so AB = I .

d) The columns of A span Rk .

19. Let A be a 4× 4 matrix with determinant 7. Give a proof or counterexample for eachof the following.

a) For some vector b the equation Ax = b has exactly one solution.

b) For some vector b the equation Ax = b has infinitely many solutions.

c) For some vector b the equation Ax = b has no solution.

d) For all vectors b the equation Ax = b has at least one solution.

20. Let A : Rn → Rk be a real matrix, not necessarily square.

a) If two rows of A are the same, show that A is not onto by finding a vector y =(y1, . . . , yk) that is not in the image of A . [Hint: This is a mental computation ifyou write out the equations Ax = y explicitly.]

b) What if A : Cn → Ck is a complex matrix?

c) More generally, if the rows of A are linearly dependent, show that it is not onto.

21. Let A : Rn → Rk be a real matrix, not necessarily square.

a) If two columns of A are the same, show that A is not one-to-one by finding a vectorx = (x1, . . . , xn) that is in the nullspace of A .

b) More generally, if the columns of A are linearly dependent, show that A is notone-to-one.

22. Let A and B be n × n matrices with AB = 0. Give a proof or counterexample foreach of the following.

a) Either A = 0 or B = 0 (or both).

5

Q11. Compute the determinant and rank of the following matrices.

a. 1 −1 20 2 40 0 0

b.

1 6 20 0 10 0 0

Q12.

Q13. Prove that (A-1)T = (AT)-1 where A is an invertible matrix.

Q14. Prove (AB)-1 = B-1A-1 using two matrices.

Q15. Using properties of determinants prove the following:

𝑦𝑧 𝑧 𝑦𝑧 𝑧𝑥 𝑥𝑦 𝑥 𝑥𝑦

= 4𝑥𝑦𝑧

Q16.

DETERMINANTS 69

(iii) A system of equations is consistent or inconsistent according as its solutionexists or not.

(iv) For a square matrix A in matrix equation AX = B

(a) If |A| ≠ 0, then there exists unique solution.

(b) If |A| = 0 and (adj A) B ≠ 0, then there exists no solution.

(c) If |A| = 0 and (adj A) B = 0, then system may or may not be consistent.

4.2 Solved Examples

Short Answer (S.A.)

Example 1 If 2 5 6 58 8 3x

x , then find x.

Solution We have 2 5 6 58 8 3x

x . This gives

2x2 – 40 = 18 – 40 ⇒ x2 = 9 ⇒ x = ± 3.

Example 2 If

2

21

2

1 1 1 11 ,

1

x x

y y yz zx xyx y zz z

Δ= Δ = , then prove that ∆ + ∆1 = 0.

Solution We have 1

1 1 1yz zx xyx y z

Interchanging rows and columns, we get

1

111

yz xzx yxy z

2

2

2

1x xyz x

y xyz yxyz

z xyz z

=

DETERMINANTS 79

2 2 2

1 1 11 cosA 1 cosB 1 cosC 0

cos A cosA cos B cosB cos C cosC

⎡ ⎤⎢ ⎥

Δ= + + + =⎢ ⎥⎢ ⎥+ + +⎣ ⎦

.

17. Find A–1 if 0 1 1

A 1 0 11 1 0

and show that 2

–1 A 3IA2 .

Long Answer (L.A.)

18. If 1 2 0

A 2 1 20 1 1

⎡ ⎤⎢ ⎥= − − −⎢ ⎥⎢ ⎥−⎣ ⎦

, find A–1.

Using A–1, solve the system of linear equations x – 2y = 10 , 2x – y – z = 8 , –2y + z = 7.

19. Using matrix method, solve the system of equations3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2 .

20. Given 2 2 4 1 1 0

A 4 2 4 , B 2 3 42 1 5 0 1 2

, find BA and use this to solve the

system of equations y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.

21. If a + b + c ≠ 0 and 0a b cb c ac a b

= , then prove that a = b = c.

22. Prove that

2 2 2

2 2 2

2 2 2

bc a ca b ab c

ca b ab c bc a

ab c bc a ca b

is divisible by a + b + c and find the

quotient.

Q17. State true or false.

a.

b.

c.

84 MATHEMATICS

47. If f (x) =

17 19 23

23 29 34

41 43 47

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

x x x

x x x

x x x

+ + +

+ + +

+ + +

= A + Bx + Cx2 + ..., then

A = ________.State True or False for the statements of the following Exercises:

48. –13A = 31A , where A is a square matrix and |A| ≠ 0.

49. (aA)–1 = –11 A

a , where a is any real number and A is a square matrix.

50. |A–1| ≠ |A|–1 , where A is non-singular matrix.

51. If A and B are matrices of order 3 and |A| = 5, |B| = 3, then|3AB| = 27 × 5 × 3 = 405.

52. If the value of a third order determinant is 12, then the value of the determinantformed by replacing each element by its co-factor will be 144.

53.1 22 3 03 4

x x x ax x x bx x x c

+ + ++ + + =+ + +

, where a, b, c are in A.P.

54. |adj. A| = |A|2 , where A is a square matrix of order two.

55. The determinant sin A cos A sin A +cosBsin B cos A sin B+cosBsin C cos A sin C+cosB

is equal to zero.

56. If the determinant +

x a p u l fy b q v m gz c r w n h

splits into exactly K determinants of

order 3, each element of which contains only one term, then the value of K is 8.

84 MATHEMATICS

47. If f (x) =

17 19 23

23 29 34

41 43 47

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

x x x

x x x

x x x

+ + +

+ + +

+ + +

= A + Bx + Cx2 + ..., then

A = ________.State True or False for the statements of the following Exercises:

48. –13A = 31A , where A is a square matrix and |A| ≠ 0.

49. (aA)–1 = –11 A

a , where a is any real number and A is a square matrix.

50. |A–1| ≠ |A|–1 , where A is non-singular matrix.

51. If A and B are matrices of order 3 and |A| = 5, |B| = 3, then|3AB| = 27 × 5 × 3 = 405.

52. If the value of a third order determinant is 12, then the value of the determinantformed by replacing each element by its co-factor will be 144.

53.1 22 3 03 4

x x x ax x x bx x x c

+ + ++ + + =+ + +

, where a, b, c are in A.P.

54. |adj. A| = |A|2 , where A is a square matrix of order two.

55. The determinant sin A cos A sin A +cosBsin B cos A sin B+cosBsin C cos A sin C+cosB

is equal to zero.

56. If the determinant +

x a p u l fy b q v m gz c r w n h

splits into exactly K determinants of

order 3, each element of which contains only one term, then the value of K is 8.

84 MATHEMATICS

47. If f (x) =

17 19 23

23 29 34

41 43 47

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )

x x x

x x x

x x x

+ + +

+ + +

+ + +

= A + Bx + Cx2 + ..., then

A = ________.State True or False for the statements of the following Exercises:

48. –13A = 31A , where A is a square matrix and |A| ≠ 0.

49. (aA)–1 = –11 A

a , where a is any real number and A is a square matrix.

50. |A–1| ≠ |A|–1 , where A is non-singular matrix.

51. If A and B are matrices of order 3 and |A| = 5, |B| = 3, then|3AB| = 27 × 5 × 3 = 405.

52. If the value of a third order determinant is 12, then the value of the determinantformed by replacing each element by its co-factor will be 144.

53.1 22 3 03 4

x x x ax x x bx x x c

+ + ++ + + =+ + +

, where a, b, c are in A.P.

54. |adj. A| = |A|2 , where A is a square matrix of order two.

55. The determinant sin A cos A sin A +cosBsin B cos A sin B+cosBsin C cos A sin C+cosB

is equal to zero.

56. If the determinant +

x a p u l fy b q v m gz c r w n h

splits into exactly K determinants of

order 3, each element of which contains only one term, then the value of K is 8.