# Rd sharma solution Chapter 5 Algebra of matrices

#### EXERCISE 5.1

Question 1

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

Sol:

Order of matrix = number of rows ​ number of column

OR

Total number of elements = number of rows number of column

Thus, to find all possible order of a matrix having 8 elements we have to find all possible ordered pairs whose product is 8.

Possible order pairs are

(1 8) ; (8 1) ; (2 4) ; (4 2) ;

AND

The possible orders of a matrix with 5 total elements are

(1 5) ; (5 1)

Question 2

2.If  and then find

(i)    (ii)

Sol :

(i)

where,

and

= 4 + (-3)

= 1

(ii)

where,

3.Let A be a matrix of order 3​​4.If ​​denotes the first row of A and ​​denotes second column, then determine the orders of matrices ​​ and ​​.

4.Construct a 2​​3 matrix whose elements ​​are given by:

(i)​

(ii)​

(iii)​

(iv)​

5.Construct a 2 ​​2 matrix whose elements ​​are given by:

(i)

(ii)

(iii)​​

(iv)

(v)

(vi)

6.Construct a 3 ​​ 4 matrix A=[​​] whose elements ​​are given by:

(i)

(ii)

(iii)

(iv)

(v)

7.Construct a matrix 4​​3 matrix whose elements are

(i)​

(ii)​

(iii)​

8.Find x,y,a and b if ​

9.Find x,y,a and b if

(i)

10.Find the value of a,b,c and d from the following equations:

11.For what values of x and y are the following matrices equal ?

​,

12.Find x,y and z so that A=B,where

​,

13.If ​,find x,y,z,w.

14.If ​​,find x,y,z,w.

15.Find the values of x and y if

16.If

Obtain the value of a,b,c,x,y and z.

17.Give an example of
(i)a row matrix which is also a column matrix,
(ii)a diagonal matrix which is not scalar,
(iii)a triangular matrix

18.The sale figure of two car dealers during January 2007 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars,while dealer B sold 7 deluxe, 2 premium and 3 standard cars.Total sales over the 2 month periods of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars.In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars.Write 2​​3 matrices summarizing sales data for January and 2 month periods for each dealer.

#### EXERCISE 5.2

1.Compute the following sums:

(i)​

(ii)

2.Let A=​​, B=​​ and C=​​.Find each of the following:

(i)2A-3B

(ii)B-4C

(iii)3A-C

(iv)3A-2B+3C

3.If A=​​,B=​​, C=​,find :

(i)A+B and B+C

(ii)2B+3A and 3C-4B

4.Let A=​​, B=​​ and C=​​. Compute 2A-3B+4C.

5.If A=diag(2 -5 9), B=diag(1 1 -4) and C=diag(-6 3 4) , find:

(i) A-2B

(ii) B+C-2A

(iii) 2A+3B-5C

6. Given the matrices

A=​​, B=​​ and C=​

Verify that (A+B)+C=A+(B+C).

7.Find the matrices X and Y, if X+Y=​​ and X-Y=​

8.Find X if Y=​​ and 2X+Y=​

9.Find the matrices X and Y , if 2X-Y=​​ and X+2Y=​

10.If X-Y=​​ and X+Y=​​ , find X and Y.

11.Find the matrix A, if ​​+A=​

12.If A=​​ , B=​​, find matrix C such that 5A+3B+2C is a null matrix.

13.If A=​​, B=​​, find matrix X such that 2A+3X=5B.

14.If A=​​ and B=​​, find the matrix C such that A+B+C is zero matrix.

15.Find x, y satisfying the matrix equations:

(i)

(ii)

16.If ​

​,find x and y.

17.Find the value of ​​, a non-zero scalar , if

18. (i) Find a matrix X such that 2A+B+X=O, where

A=​​, B=​

(ii) If A=​​ and B=​​, then find the matrix X of order 3​​2 such that 2A+3X=5B.

19.Find x, y, z and t, if

(i)

(ii)

20.In a certain city there are 30 colleges. Each colleges has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix . Using scalar multiplication, find the total number of posts of each kind in all the colleges.

#### EXERCISE 5.3

1.Compute the indicated products:

(i) ​

(ii) ​

(iii)

2. Show that AB​​BA in each of the following cases:

(i) A=​​ and  B=​

(ii) A=​​ and B=​

(iii) A=​​ and  B=​

3.Compare the products AB and BA whichever exists in each of the following cases:

(i) A=​​ and B=​

(ii) A=​​ and  B=​

(iii) A=​​ and  B=​

(iv)

4. Show that AB​​BA in each of the following cases:

(i) A=​​ and  B=​

(ii) A=​​ and B=​

5. Evaluate the following:

(i)

(ii)

(iii)

6. If A=​​, B=​​ and C=​,then show that ​

7. If A=​​ and B=​​,find ​

8. If A=​​,prove that (A-2I)(A-3I)=0

9. If A=​​,show that ​​and ​

10. If A=​​,show that ​

11. If A=​​, find ​

12. If A=​​ and B=​​show that Ab=BA=​

13. If A=​​ and B=​​show that AB=BA=​

14. If A=​​ and B=​​, show that AB=A and BA=B.

15. Let A=​​ and B=​​, compute ​

16. For the following matrices verify the associativity of matrix multiplicationi.e. (AB)C=A(BC):

(i) A=, B=​​ and C=​

(ii) A=​​, B=​​ , C=​

17. For the following matrices verify the distributivity of the matrix multiplication over matrix addition i.e. A(B+C)=AB+AC

(i) A=​​ , B=​​ , C=​

(ii) A=​​ , B=​​ , C=​

18. If A=​​ , B=​​ , C=​

19.Compute the following elements ​​ and ​​ of the matrix :

A=​

20. If A=, and I is the identity matrix of order 3 , show that

21. If ​​ is a complex cube root of unity, show that

22. If  A=, show that ​

23. If A=​​, show that ​

24. If  ​​ =0,find x.

25. If  ​​ =0, find x.

26. If A=

27. If A=

28. If A=

29. If A=

30. If A=

31. Show that the matrix A=

32. Show that the matrix A=

33.If A=

34.

(i) If A=

(ii) If A=

35. If A=

36. If A=

37. If A=

38. If A=

39.

40.

41. If A=

42. If A=

43. If A=

44. If A=

45. If A=

46. If A=

47.With using the concept of inverse matrix, find the matrix ​​ such that

48.

49.

50. If A=

51. If A=

52. If A=

53. If A=

54. If A=

55. Let A=

56. If B, C are n rowed square matrices and if A=B+C , BC=CB , ​​ , then show that for every

57. If A=diag(a,b,c) show that  for all positive integers n .

58. A matrix X has a+b rows and a+2 columns while the matrix Y has b+1 rows and a+3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

59. Give examples of matrices

(i) A and B such that

(ii) A and B such that AB=0 but​

(iii) A and B such that AB=0 but

(iv) A and B and C such that AB=AC but

60. Let A and B be square matrices of the same order. Does hold ? If not, why ?

61. If A and B are square matrices of the same order, explain, why in general

(i)

(ii)

(iii)

62. Three shopkeeper A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks,  dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebook costs 40 paise, a pen costs ₹ 1.25 and  a pencil costs 35 paise. Use matrix multiplication to calculate each individual’s bill.

63. The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are ₹ 8.30 , ₹ 3.45 , and ₹ 4.50 each respectively. Find the total amount the store will receive from selling all items.

64. In legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways : telephone , house calls and letters. The cost per contact (in paise) is given in matrix A as

A =

The number of contacts of each type made in two cities X and Y is given in matrix B as :

B =

Find the total amount spent by the group in the two cities X and Y.

65. A trust fund has ₹ 30,000 that must be invested in two different types of bonds.The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication determine bow to divide ₹ 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of

(i) ₹ 1800

(ii) ₹ 2000

66. Let A=​​, B=​​ and C=​​. Verify that AB=AC though B​​C , A​​0

# ALGEBRA OF MATRICES

#### EXERCISE 5.1

1.If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

solution:

Order of matrix=number of rows ​​number of column

or

Total number of elements=number of rows ​​number of column

Thus, to find all possible order of a matrix having 8 elements we have to find all possible ordered pairs whose product is 8.

Possible order pairs are

(18) ; (81) ; (2​​4) ; (42) ;

AND

The possible orders of a matrix with 5 total elements are

(1  5) ; (5  1)

2.If

then find

(i)​ ​(ii)​

solution:

(i)

where,

​and ​

(ii)

where,

3.Let A be a matrix of order 3​​4.If ​​denotes the first row of A and ​​denotes second column, then determine the orders of matrices ​​ and ​​.

4.Construct a 2​​3 matrix whose elements ​​are given by:

(i)​

(ii)​

(iii)​

(iv)​

5.Construct a 2 ​​2 matrix whose elements ​​are given by:

(i)

(ii)

(iii)​​

(iv)

(v)

(vi)

6.Construct a 3 ​​ 4 matrix A=[​​] whose elements ​​are given by:

(i)

(ii)

(iii)

(iv)

(v)

7.Construct a matrix 4​​3 matrix whose elements are

(i)​

(ii)​

(iii)​

8.Find x,y,a and b if ​

9.Find x,y,a and b if

(i)

10.Find the value of a,b,c and d from the following equations:

11.For what values of x and y are the following matrices equal ?

​,

12.Find x,y and z so that A=B,where

​,

13.If ​,find x,y,z,w.

14.If ​​,find x,y,z,w.

15.Find the values of x and y if

16.If

Obtain the value of a,b,c,x,y and z.

17.Give an example of
(i)a row matrix which is also a column matrix,
(ii)a diagonal matrix which is not scalar,
(iii)a triangular matrix

18.The sale figure of two car dealers during January 2007 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars,while dealer B sold 7 deluxe, 2 premium and 3 standard cars.Total sales over the 2 month periods of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars.In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars.Write 2​​3 matrices summarizing sales data for January and 2 month periods for each dealer.

#### EXERCISE 5.2

1.Compute the following sums:

(i)​

(ii)

2.Let A=​​, B=​​ and C=​​.Find each of the following:

(i)2A-3B

(ii)B-4C

(iii)3A-C

(iv)3A-2B+3C

3.If A=​​,B=​​, C=​,find :

(i)A+B and B+C

(ii)2B+3A and 3C-4B

4.Let A=​​, B=​​ and C=​​. Compute 2A-3B+4C.

5.If A=diag(2 -5 9), B=diag(1 1 -4) and C=diag(-6 3 4) , find:

(i) A-2B

(ii) B+C-2A

(iii) 2A+3B-5C

6. Given the matrices

A=​​, B=​​ and C=​

Verify that (A+B)+C=A+(B+C).

7.Find the matrices X and Y, if X+Y=​​ and X-Y=​

8.Find X if Y=​​ and 2X+Y=​

9.Find the matrices X and Y , if 2X-Y=​​ and X+2Y=​

10.If X-Y=​​ and X+Y=​​ , find X and Y.

11.Find the matrix A, if ​​+A=​

12.If A=​​ , B=​​, find matrix C such that 5A+3B+2C is a null matrix.

13.If A=​​, B=​​, find matrix X such that 2A+3X=5B.

14.If A=​​ and B=​​, find the matrix C such that A+B+C is zero matrix.

15.Find x, y satisfying the matrix equations:

(i)

(ii)

16.If ​

​,find x and y.

17.Find the value of ​​, a non-zero scalar , if

18. (i) Find a matrix X such that 2A+B+X=O, where

A=​​, B=​

(ii) If A=​​ and B=​​, then find the matrix X of order 3​​2 such that 2A+3X=5B.

19.Find x, y, z and t, if

(i)

(ii)

20.In a certain city there are 30 colleges. Each colleges has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix . Using scalar multiplication, find the total number of posts of each kind in all the colleges.

#### EXERCISE 5.3

1.Compute the indicated products:

(i) ​

(ii) ​

(iii)

2. Show that AB​​BA in each of the following cases:

(i) A=​​ and  B=​

(ii) A=​​ and B=​

(iii) A=​​ and  B=​

3.Compare the products AB and BA whichever exists in each of the following cases:

(i) A=​​ and B=​

(ii) A=​​ and  B=​

(iii) A=​​ and  B=​

(iv)

4. Show that AB​​BA in each of the following cases:

(i) A=​​ and  B=​

(ii) A=​​ and B=​

5. Evaluate the following:

(i)

(ii)

(iii)

6. If A=​​, B=​​ and C=​,then show that ​

7. If A=​​ and B=​​,find ​

8. If A=​​,prove that (A-2I)(A-3I)=0

9. If A=​​,show that ​​and ​

10. If A=​​,show that ​

11. If A=​​, find ​

12. If A=​​ and B=​​show that Ab=BA=​

13. If A=​​ and B=​​show that AB=BA=​

14. If A=​​ and B=​​, show that AB=A and BA=B.

15. Let A=​​ and B=​​, compute ​

16. For the following matrices verify the associativity of matrix multiplicationi.e. (AB)C=A(BC):

(i) A=, B=​​ and C=​

(ii) A=​​, B=​​ , C=​

17. For the following matrices verify the distributivity of the matrix multiplication over matrix addition i.e. A(B+C)=AB+AC

(i) A=​​ , B=​​ , C=​

(ii) A=​​ , B=​​ , C=​

18. If A=​​ , B=​​ , C=​

19.Compute the following elements ​​ and ​​ of the matrix :

A=​

20. If A=, and I is the identity matrix of order 3 , show that

21. If ​​ is a complex cube root of unity, show that

22. If  A=, show that ​

23. If A=​​, show that ​

24. If  ​​ =0,find x.

25. If  ​​ =0, find x.

26. If A=

27. If A=

28. If A=

29. If A=

30. If A=

31. Show that the matrix A=

32. Show that the matrix A=

33.If A=

34.

(i) If A=

(ii) If A=

35. If A=

36. If A=

37. If A=

38. If A=

39.

40.

41. If A=

42. If A=

43. If A=

44. If A=

45. If A=

46. If A=

47.With using the concept of inverse matrix, find the matrix ​​ such that

48.

49.

50. If A=

51. If A=

52. If A=

53. If A=

54. If A=

55. Let A=

56. If B, C are n rowed square matrices and if A=B+C , BC=CB , ​​ , then show that for every

57. If A=diag(a,b,c) show that ​​ for all positive integers n .

58. A matrix X has a+b rows and a+2 columns while the matrix Y has b+1 rows and a+3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

59. Give examples of matrices

(i) A and B such that ​

(ii) A and B such that AB=0 but ​

(iii) A and B such that AB=0 but ​

(iv) A and B and C such that AB=AC but ​

60. Let A and B be square matrices of the same order. Doeshold ? If not, why ?

61. If A and B are square matrices of the same order, explain, why in general

(i) ​

(ii) ​

(iii) ​

62. Three shopkeeper A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks,  dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebook costs 40 paise, a pen costs ₹ 1.25 and  a pencil costs 35 paise. Use matrix multiplication to calculate each individual’s bill.

63. The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are ₹ 8.30 , ₹ 3.45 , and ₹ 4.50 each respectively. Find the total amount the store will receive from selling all items.

64. In legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways : telephone , house calls and letters. The cost per contact (in paise) is given in matrix A as

A=​

The number of contacts of each type made in two cities X and Y is given in matrix B as :

B=​

Find the total amount spent by the group in the two cities X and Y.

65. A trust fund has ₹ 30,000 that must be invested in two different types of bonds.The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication determine bow to divide ₹ 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of

(i) ₹ 1800

(ii) ₹ 2000

66. Let A=​​, B=​​ and C=​​. Verify that AB=AC though B​​C , A​​0