Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 1 - Matrices and Determinants [Latest edition]

#### Chapters ## Chapter 1: Matrices and Determinants

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Miscellaneous Problems
Exercise 1.1 [Pages 7 - 8]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 1 Matrices and Determinants Exercise 1.1 [Pages 7 - 8]

Exercise 1.1 | Q 1. (i) | Page 7

Find the minors and cofactors of all the elements of the following determinant.

|(5,20),(0, -1)|

Exercise 1.1 | Q 1. (ii) | Page 7

Find the minors and cofactors of all the elements of the following determinant.

|(1,-3,2),(4,-1,2),(3,5,2)|

Exercise 1.1 | Q 2. | Page 7

Evaluate |(3,-2,4),(2,0,1),(1,2,3)|

Exercise 1.1 | Q 3. | Page 7

Solve: |(2,x,3),(4,1,6),(1,2,7)| = 0

Exercise 1.1 | Q 4. | Page 7

Find |AB| if A = [(3,-1),(2,1)] and B = [(3,0),(1,-2)]

Exercise 1.1 | Q 5. | Page 7

Solve: |(7,4,11),(-3,5,x),(-x,3,1)| = 0

Exercise 1.1 | Q 6. | Page 7

Evaluate: |(1,a,a^2 - bc),(1,b,b^2 - ca),(1,c,c^2 - ab)|

Exercise 1.1 | Q 7. | Page 8

Prove that |(1/a,bc,b+c),(1/b,ca,c+a),(1/c,ab,a+b)| = 0

Exercise 1.1 | Q 8. | Page 8

Prove that |(-a^2,ab,ac),(ab,-b^2,bc),(ac,bc,-c^2)| = 4a^2b^2c^2

Exercise 1.2 [Page 12]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 1 Matrices and Determinants Exercise 1.2 [Page 12]

Exercise 1.2 | Q 1. | Page 12

Find the adjoint of the matrix A = [(2,3),(1,4)]

Exercise 1.2 | Q 2. | Page 12

If A = [(1,3,3),(1,4,3),(1,3,4)] then verify that A(adj A) = |A| I and also find A-1.

Exercise 1.2 | Q 3. (i) | Page 12

Find the inverse of the following matrix:

[(1,-1),(2,3)]

Exercise 1.2 | Q 3. (ii) | Page 12

Find the inverse of the following matrix:

[(3,1),(-1,3)]

Exercise 1.2 | Q 3. (iii) | Page 12

Find the inverse of the following matrix:

[(1,2,3),(0,2,4),(0,0,5)]

Exercise 1.2 | Q 3. (iv) | Page 12

Find the inverse of the following matrix:

[(-3,-5,4),(-2,3,-1),(1,-4,-6)]

Exercise 1.2 | Q 4. | Page 12

If A = [(2,3),(1,-6)] and B = [(-1,4),(1,-2)], then verify adj (AB) = (adj B)(adj A)

Exercise 1.2 | Q 5. | Page 12

If A = [(2,-2,2),(2,3,0),(9,1,5)] then, show that (adj A) A = O.

Exercise 1.2 | Q 6. | Page 12

If A = [(-1,2,-2),(4,-3,4),(4,-4,5)]  then, show that the inverse of A is A itself.

Exercise 1.2 | Q 7. | Page 12

If A-1 = [(1,0,3),(2,1,-1),(1,-1,1)]  then, find A.

Exercise 1.2 | Q 8. | Page 12

Show that the matrices A = [(2,2,1),(1,3,1),(1,2,2)] and B = [(4/5,(-2)/5,(-1)/5),((-1)/5,3/5,(-1)/5),((-1)/5,(-2)/5,4/5)] are inverses of each other.

Exercise 1.2 | Q 9. | Page 12

If A = [(3,7),(2,5)] and B = [(6,8),(7,9)], then verify that (AB)-1 = B-1A-1

Exercise 1.2 | Q 10. | Page 12

Find m if the matrix [(1,1,3),(2,λ,4),(9,7,11)] has no inverse.

Exercise 1.2 | Q 11. | Page 12

If X = [(8,-1,-3),(-5,1,2),(10,-1,-4)] and Y = [(2,1,-1),(0,2,1),(5,p,q)]  then, find p, q if Y = X-1

Exercise 1.3 [Page 15]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 1 Matrices and Determinants Exercise 1.3 [Page 15]

Exercise 1.3 | Q 1. | Page 15

Solve by matrix inversion method:

2x + 3y – 5 = 0; x – 2y + 1 = 0.

Exercise 1.3 | Q 2. (i) | Page 15

Solve by matrix inversion method:

3x – y + 2z = 13; 2x + y – z = 3; x + 3y – 5z = - 8

Exercise 1.3 | Q 2. (ii) | Page 15

Solve by matrix inversion method:

x – y + 2z = 3; 2x + z = 1; 3x + 2y + z = 4

Exercise 1.3 | Q 2. (iii) | Page 15

Solve by matrix inversion method:

2x – z = 0; 5x + y = 4; y + 3z = 5

Exercise 1.3 | Q 3. | Page 15

A sales person Ravi has the following record of sales for the month of January, February and March 2009 for three products A, B and C. He has been paid a commission at fixed rate per unit but at varying rates for products A, B and C.

 Months Sales in units Commission A B C January 9 10 2 800 February 15 5 4 900 March 6 10 3 850

Find the rate of commission payable on A, B and C per unit sold using matrix inversion method.

Exercise 1.3 | Q 4. | Page 15

The prices of three commodities A, B, and C are ₹ x, y, and z per unit respectively. P purchases 4 units of C and sells 3 units of A and 5 units of B. Q purchases 3 units of B and sells 2 units of A and 1 unit of C. R purchases 1 unit of A and sells 4 units of B and 6 units of C. In the process P, Q and R earn ₹ 6,000, ₹ 5,000 and ₹ 13,000 respectively. By using the matrix inversion method, find the prices per unit of A, B, and C.

Exercise 1.3 | Q 5. | Page 15

The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using the matrix inversion method find the numbers.

Exercise 1.3 | Q 6. | Page 15

Weekly expenditure in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using the matrix inversion method.

 Week Number of employees Total weekly salary(in ₹) A B C 1st week 4 2 3 4900 2nd week 3 3 2 4500 3rd week 4 3 4 5800
Exercise 1.4 [Pages 19 - 20]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 1 Matrices and Determinants Exercise 1.4 [Pages 19 - 20]

Exercise 1.4 | Q 1 | Page 19

The technology matrix of an economic system of two industries is |(0.50,0.30),(0.41,0.33)| Test whether the system is viable as per Hawkins Simon conditions.

Exercise 1.4 | Q 2 | Page 19

The technology matrix of an economic system of two industries is |(0.6,0.9),(0.20,0.80)|.
Test whether the system is viable as per Hawkins-Simon conditions.

Exercise 1.4 | Q 3 | Page 19

The technology matrix of an economic system of two industries is |(0.50,0.25),(0.40,0.67)|. Test whether the system is viable as per Hawkins-Simon conditions.

Exercise 1.4 | Q 4 | Page 19

Two commodities A and B are produced such that 0.4 tonne of A and 0.7 tonne of B are required to produce a tonne of A. Similarly 0.1 tonne of A and 0.7 tonne of B are needed to produce a tonne of B. Write down the technology matrix. If 68 tonnes of A and 10.2 tonnes of B are required, find the gross production of both of them.

Exercise 1.4 | Q 5 | Page 20

Suppose the inter-industry flow of the product of two industries are given as under.

 Production sector Consumption sector Domestic demand Total output X Y X 30 40 50 120 Y 20 10 30 60

Determine the technology matrix and test Hawkin’s -Simon conditions for the viability of the system. If the domestic demand changes to 80 and 40 units respectively, what should be the gross output of each sector in order to meet the new demands.

Exercise 1.4 | Q 6 | Page 20

You are given the following transaction matrix for a two-sector economy.

 Sector Sales Final demand Gross output 1 2 1 4 3 13 20 2 5 4 3 12
1. Write the technology matrix
2. Determine the output when the final demand for the output sector 1 alone increases to 23 units.
Exercise 1.4 | Q 7 | Page 20

Suppose the inter-industry flow of the product of two sectors X and Y are given as under.

 Production Sector Consumption sector Domestic demand Gross output X Y X 15 10 10 35 Y 20 30 15 65

Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

Exercise 1.5 [Pages 20 - 22]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 1 Matrices and Determinants Exercise 1.5 [Pages 20 - 22]

Exercise 1.5 | Q 1 | Page 20

The value of x if |(0,1,0),(x,2,x),(1,3,x)| = 0 is

• 0, –1

• 0, 1

• –1, 1

• –1, –1

Exercise 1.5 | Q 2 | Page 20

The value of |(2x + y,x,y),(2y+z,y,z),(2z+x,z,x)| is

• xyz

• x + y + z

• 2x + 2y + 2z

• 0

Exercise 1.5 | Q 3 | Page 20

The cofactor of –7 in the determinant |(2,-3,5),(6,0,4),(1,5,-7)| is

• -18

• 18

• -7

• 7

Exercise 1.5 | Q 4 | Page 20

If Delta = |(1,2,3),(3,1,2),(2,3,1)| then |(3,1,2),(1,2,3),(2,3,1)| is

• Δ

• -3Δ

Exercise 1.5 | Q 5 | Page 20

The value of the determinant [(a,0,0),(0,b,0),(0,0,c)]^2 is

• abc

• 0

• a2b2c2

• - abc

Exercise 1.5 | Q 6 | Page 21

If A is square matrix of order 3 then |kA| is

• k|A|

• - k|A|

• k3|A|

• - k3|A|

Exercise 1.5 | Q 7 | Page 21

Exercise 1.5 | Q 8 | Page 21

The inverse matrix of ((4/5,(-5)/12),((-2)/5,1/2)) is

• 7/30 ((1/2,5/12),(2/5,4/5))

• 7/30 ((1/2,(-5)/12),((-2)/5,1/5))

• 30/7 ((1/2,5/12),(2/5,4/5))

• 30/7 ((1/2,(-5)/12),((-2)/5,4/5))

Exercise 1.5 | Q 9 | Page 21

If A = [(a,b),(c,d)] such that ad - bc ≠ 0 then A-1 is

• 1/("ad" - "bc") ((d,b),(-c,a))

• 1/("ad" - "bc") ((d,b),(c,a))

• 1/("ad" - "bc") ((d,-b),(-c,a))

• 1/("ad" - "bc") ((d,-b),(c,a))

Exercise 1.5 | Q 10 | Page 21

The number of Hawkins-Simon conditions for the viability of an input-output analysis is ______.

• 1

• 3

• 4

• 2

Exercise 1.5 | Q 11 | Page 21

The inventor of input-output analysis is ______.

• Sir Francis Galton

• Fisher

• Prof. Wassily W. Leontief

• Arthur Caylay

Exercise 1.5 | Q 12 | Page 21

Which of the following matrix has no inverse

• ((-1,1),(1,-4))

• ((2,-1),(-4,2))

• ((cos a, sin a),(-sin a, cos a))

• ((sin a, sin a),(-cos a, cos a))

Exercise 1.5 | Q 13 | Page 21

The inverse matrix of ((3,1),(5,2)) is

• ((2,-1),(-5,3))

• ((-2,5),(1,-3))

• ((3,-1),(-5,-3))

• ((-3,5),(1,-2))

Exercise 1.5 | Q 14 | Page 21

If A = ((-1,2),(1,-4)) then A(adj A) is

• ((-4,-2),(-1,-1))

• ((4,-2),(-1,1))

• ((2,0),(0,2))

• ((0,2),(2,0))

Exercise 1.5 | Q 15 | Page 21

If A and B non-singular matrix then, which of the following is incorrect?

• A2 = I implies A-1 = A

• I-1 = I

• If AX = B then X = B-1A

• If A is square matrix of order 3 then |adj A| = |A|2

Exercise 1.5 | Q 16 | Page 21

The value of |(5,5,5),(4x,4y,4z),(-3x,-3y,-3z)| is

• 5

• 4

• 0

• -3

Exercise 1.5 | Q 17 | Page 21

If A is an invertible matrix of order 2 then det (A-1) be equal

• det(A)

• 1/(det("A"))

• 1

• 0

Exercise 1.5 | Q 18 | Page 22

If A is 3 × 3 matrix and |A| = 4 then |A-1| is equal to:

• 1/4

• 1/16

• 2

• 4

Exercise 1.5 | Q 19 | Page 22

If A is a square matrix of order 3 and |A| = 3 then |adj A| is equal to:

• 81

• 27

• 3

• 9

Exercise 1.5 | Q 20 | Page 22

The value of |(x,x^2 - yz,1),(y,y^2-zx,1),(z,z^2-xy,1)| is

• 1

• 0

• -1

• -xyz

Exercise 1.5 | Q 21 | Page 22

If A = |(cos theta,sin theta),(-sin theta,cos theta)|, then |2A| is equal to

• 4 cos 2θ

• 4

• 2

• 1

Exercise 1.5 | Q 22 | Page 22

If Δ = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|  and Aij is cofactor of aij, then value of Δ is given by:

• a11A31 + a12A32 + a13A33

• a11A11 + a12A21 + a13A31

• a21A11 + a22A12 + a23A13

• a11A11 + a21A21 + a31A31

Exercise 1.5 | Q 23 | Page 22

If |(x,2),(8,5)| = 0 then the value of x is

• (-5)/6

• 5/6

• (-16)/5

• 16/5

Exercise 1.5 | Q 24 | Page 22

If |(4,3),(3,1)| = -5 then the value of |(20,15),(15,5)| is:

• -5

• -125

• -25

• 0

Exercise 1.5 | Q 25 | Page 22

If any three rows or columns of a determinant are identical then the value of the determinant is:

• 0

• 2

• 1

• 3

Miscellaneous Problems [Pages 22 - 23]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 1 Matrices and Determinants Miscellaneous Problems [Pages 22 - 23]

Miscellaneous Problems | Q 1 | Page 22

Solve: |(x,2,-1),(2,5,x),(-1,2,x)| = 0

Miscellaneous Problems | Q 2 | Page 22

Evaluate |(10041,10042,10043),(10045,10046,10047),(10049,10050,10051)|

Miscellaneous Problems | Q 3 | Page 22

Without actual expansion show that the value of the determinant |(5,5^2,5^3),(5^2,5^3,5^4),(5^4,5^5,5^6)| is zero.

Miscellaneous Problems | Q 4 | Page 22

Show that |(0,ab^2,ac^2),(a^2b,0,bc^2),(a^2c,b^2c,0)| = 2a^3b^3c^3

Miscellaneous Problems | Q 5 | Page 22

If A = |(1,1,1),(3,4,7),(1,-1,1)| verify that A(adj A) = (adj A)(A) = |A|I3.

Miscellaneous Problems | Q 6 | Page 22

If A = |(3,-1,1),(-15,6,-5),(5,-2,2)| then, find the Inverse of A.

Miscellaneous Problems | Q 7 | Page 22

If A = [(1,-1),(2,3)] show that A2 - 4A + 5I2 = 0 and also find A-1.

Miscellaneous Problems | Q 8 | Page 22

Solve by using matrix inversion method:

x - y + z = 2, 2x - y = 0, 2y - z = 1

Miscellaneous Problems | Q 9 | Page 22

The cost of 2 Kg of Wheat and 1 Kg of Sugar is ₹ 70. The cost of 1 Kg of Wheat and 1 Kg of Rice is ₹ 70. The cost of 3 Kg of Wheat, 2 Kg of Sugar and 1 Kg of rice is ₹ 170. Find the cost of per kg each item using the matrix inversion method.

Miscellaneous Problems | Q 10 | Page 23

The data are about an economy of two industries A and B. The values are in crores of rupees.

 Producer User Final demand Total outout A B A 50 75 75 200 B 100 50 50 200

Find the output when the final demand changes to 300 for A and 600 for B.

## Chapter 1: Matrices and Determinants

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Miscellaneous Problems ## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 1 - Matrices and Determinants

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Concepts covered in Class 11th Business Mathematics and Statistics Answers Guide chapter 1 Matrices and Determinants are Determinants, Input–Output Analysis, Inverse of Matrix.

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