**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 34.If denotes the first row of A and denotes second column, then determine the orders of matrices and .**

**4.Construct a 23 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 43 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 23 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 32 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 ABBA 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 ABBA 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 BC , A0

__ALGEBRA OF MATRICES__

__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) ; (24) ; (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 34.If denotes the first row of A and denotes second column, then determine the orders of matrices and .**

**4.Construct a 23 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 43 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 23 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 32 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 ABBA 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 ABBA 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 BC , A0