reliable.redik.inreliable.redik.in/uploads/documents/Matrices_-_2.doc · Web viewQ50. 2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1 (B) By Method of Inversion For solving

CHAPTER - 2 MATRICES

Marks allotted: 8

Definition : MatrixA matrix is an ordered rectangular array of numbers or functions enclosed in brackets,

[ ] or ( ).The numbers or the functions forming a matrix are called as the elements of a matrix.A matrix is denoted by a capital letter. The following are some matrices.

C= [cos - sin tan ]

Definition: Order of a matrix.The order of a matrix having m rows and n columns is m x n or we can say it is a matrix of order m x n (You read this as order m by n).

Thus, if we refer to the above matrices, we can say that the matrix A is of order 2 X 3, matrix B is of order 2 X 2 and matrix C is of order 1 X 3.

In general, a matrix of order m x n has the following arrangement.A =

In this arrangement, the first row or R1 is given as a11, a12,…. a1nand the first column or C1 is given as

Any element of matrix A is denoted by aij, where ‘i’ indicates the row and ‘j’ indicated the column to which the element belongs.

Therefore, more compact notation which can be used for the above matrix isA = [aij]m x n where i = 1,2,….m

And j = 1,2,…..nAs each row of matrix A has n elements and there ‘m’ rows in all, the number of elements in the above matrix is equal to mn.

Types of matrices1. Row matrix: A matrix having only one row is called a row matrix. Its order is 1 x n.

e.g. [ 4 ], [1 2 ], [ 0 -1 3], [p q r s] are all row matrices.

2. Column matrix : A matrix having only one column is called as a column matrix. Its order is

m x 1.

e.g. [-5], , , are all column matrices.

3. Square matrix : A matrix in which the number of rows and columns are equal, is called a square matrix.A square matrix having m rows and m column is said to be order m.

e.g. A [1] 1x1 , B = C = are square matrices.

4. Zero matrix: A matrix is called as a zero matrix or a null matrix if all of its elements are zero.

e.g [0],[ 0 0],

are all zero matrices. A zero matrix is denoted by O.15

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5. Diagonal matrix: A square matrix A = [a ij] m x m is called a diagonal matrix if a ij = 0 for all i ≠ j.

e.g. [2], are all diagonal matrices.

6. Scalar matrix: A diagonal matrix A = [aij] m x m is called a scalar matrix if all the diagonal elements are equal.

e.g. are scalar matrices.

7. Identity matrix: A scalar matrix A = [aij] m x m is called an identity or a unit matrix if all the diagonal elements are unity.

e.g. [ 1 ],

An identity matrix is denoted by . If its order is 2, it is denoted by 2. If its order is 3, it is denoted by 3 etc.

8. Transpose of a matrix:If A = [aij] is an m x n matrix then the matrix obtained by interchanging the rows and the columns of A is called as the transpose of A. It is denoted as A’ or A T. It is clear that if A = [aij] is of order m x n then A’ = [aji ] is of order n x m.

e.g. A = A =

B = B =

9. Symmetric matrix: A square matrix A = [aij] m x m is called a symmetric matrix if aij = aji for all i,jNote that A = [aij] m x m is symmetric if and only if A = A’.

e.g. A = is a symmetric matrix because a12 = a21.

B = is a symmetric matrix because a12 = a21 , a13 = a31 and a23 = a32.

10. Skew – Symmetric Matrix: A square matrix A = [aij] m x m is a skew symmetric matrix if aij = -aji for all i and jNote: A is a skew symmetric matrix if and only A = -A’

A= A = , A=

11. Upper – triangular matrix: A square matrix A = [aij]m x m is called an upper – triangular matrix if aij = 0 when i > j

e.g. is an upper – triangular matrix.

12. Lower – triangular matrix : A square matrix A = [aij] m x m is called a lower –triangular matrix if aij = 0 when i < j.

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e.g. is a lower – triangular matrix.

13. A triangular matrix: A matrix, which is either upper – triangular or a lower –triangular matrix, is called a triangular matrix.

e.g. and are both triangular matrices.

14. Singular matrix : A square matrix A is called a singular matrix, if l A l = 0

For example, if A = then l A l = 0 Hence A is a singular matrix.

15. Non-singular matrix : A square matrix A is called a non-singular matrix, if l A l ≠ 0.

For example, if A = , then l A l = - 5 ≠ 0. Hence A is a non – singular matrix.

Type 1 : Using definitions & types of matricesQ1 (i) State the order and the type of each of the following matrices.

(a) (b) (c)

(d) (e) (f)

(ii) Write the elements a11, a31, a33,a24 and a23 (whichever exists) for the following matrices.

(a) (b) (c)

(d) (e) (f)

Q2. If a matrix contains 9 elements, then write all the possible orders it can have.

Q3. (i) Find a matrix A = such that for all i and j.

(ii) Find a matrix A = such that for all i and j.

Q4. Write the transpose of the following matrices.(i) (ii)

(ii) (iii) (v)

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Q5. State the type of the matrix A = Find A’ What is your observation? What

can you con clued?Q6. (i) Give examples of two matrices of order 2 x 2 which are symmetric.

(ii) Give examples of two matrices of order 3 x 3 which are symmetric.

Q7. (i) Give examples of two matrices of order 2 x 2 which are skew-symmetric.(ii) Give examples of two matrices of order 3 x 3 which are skew-symmetric.

Q8. Determine whether the following matrices are singular or non-singular

(i) (ii) (iii)

(iv) (v) (vi)

(vii) Find k so that A is a singular matrix where A =

Algebra of Matrices.1. Equality of Matrices.

Definition: Two matrices A = [aij] and B = [bij] are said to be equal if(a) They are of the same order and (b) Each element of A is equal to he corresponding element of B. i.e. a ij = bij for all i, j,

For example, and are both identity matrices but they are not equal.

Further, if = then a = 1, b = 2. c = 3 and d = 4.

If two matrices A and B are equal, we write A = B.2. Multiplication of a matrix by a scalar:

If A = [aij] m x n is a matrix and k is any scalar then the multiplication of A by k is a matrix kA which is of the same order m x n such that kA = [kaij] m x n for all I, j.Thus, kA is obtained by multiplying each element of A by scalar k.

For example, if A = and k = 3

then k A = 3A =

3A =

Note: 1. Scalar multiplication does not change the order of the matrix.2. If A is a given matrix then (-1) A = -A. This is called as the negative matrix of A or

simply negative of A.

3. Additional of two matrices.If A = [aij] and B = [bij] are two matrices of the same order m x n, then the additional of two matrices A and B is defined. It is a matrix C = [cj] m x n where cij = aij + bij for all i,j.

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For example, A = and B =

Then A + B = C = =

Note that A + A + A + …. (k times) is same as kA, where k N.

4. Subtraction of two matrices:For two matrices A and B of same order, A – B is defined as A – B = A + (-B) where (-B) is negative matrix of B.

Foe example, A = and B = the A – B = A + (-B)

=

=

=

Type 2 : Addition & Subtraction between two matrices.Q9. For the following matrices, find the given scalar multiplications.

(i) A = , find 2A (ii) B = , find 3B (iii) C = , find

-4C

Q10. If A = , B = and C = , find(i) A + B (ii) B – A (iii) A + B + C (iv) (C - B) - A

Q11. A = and B = C = , then find

(i) 2A – B + C (ii) A + C – 3B (iii) 2C – B + 3A

Q12. Find X and Y, if(i) X + Y = , X – Y =

(ii) 2X – 3Y = , 3X + 2Y =

(iii) Solve the equations for, P and Q, P + 2Q = and 2P – Q = Q13. Find the values of x and y, if

(i) (ii)

Q14.. Find the values of x, y, z and w, if

Q15. If A = and B = , find a matrix C such that A + B+ C is a zero matrix.

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Type 3: Multiplication between two matricesDefinition: Multiplication between two matrices is possible if pre-factor (1st factor) columns is

equal to post-factor (2nd factor) rows and the order of resultant matrix will be pre-factor (1st factor) rows X post-factor (2nd factor) columns

For example, if A = and B = are the given matrices, then the

product AB is defined and it is of order 2 x 2.

Here [A B] = (A2 x 3) x (B3x2)

=

=

=

Properties of matrix multiplication. 1. The associative law: For the matrices A, B and C,

(AB) C = A (BC)

Note that this is true only if both the sides of the above equation are defined.

2. The distributive law: For three matrices A, B and C,(a) A (B + C) = AB + AC(b) (A + B) C = AC + BC

Note that this is true only if both the sides of the equation are defined.

3. Multiplicative Identify: For every square matrix A, there is an identity matrix of the same order such that A = A = A.I is, thus, the multiplicative identity of a square matrix.

4. For two matrices A and B if AB = O, it does not necessarily mean that either A or B are zero matrices.

e.g. If A = and B = then AB = , but neither A nor B is a zero matrix.

5. If for three matrices A, B and C AB = AC, then this does not necessarily mean B = C.

6. Matrix multiplication is not necessarily commutative. For any two matrices A and B, it is possible that(a) Product AB is defined, but BA is not defined.(b) Product AB is not defined, but BA is defined.(c) AB and BA both are defined, but AB ≠ BA.(d) AB and BA both are defined and AB = BA.The positive integral power of a square matrix A is defined as An = A x A x A x ……. n

times.

Thus A2 = A A, A3 = A.A.A = A2. A A4 =A. A. A. A = A3A etc.

Important Results:If A, B are square matrices of the same order, than(i) (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 A2 + 2AB + B2.(ii) (A + B)(A – B) = A2 - AB + BA + B2 A2 – B2.(iii) If (A + B)2 = A2 + B2 then AB + BA = 0.

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(iv) If (A + B)(A – B) = A2 – B2, then AB = BA

Q16. Find the following products.

(i) (ii)

(iii) (iv)

(v) (vi)

(vii)

Q17. (i) If A = , B = and C= find AB and AC. What is your observation?

(ii) If A = , B = , verify that AB = 0 and BA 0.What conclusion can be drawn from this?

Q18. Simplify + sin

Q19. If A = and B = , C= , verify that (AB) C = A (BC)

Q20. (i) Simplify

(ii) Find x, y, z if

Q21. If A = , then show that A2 – 5A is a scalar matrix.

Q22. If A = , find A2 – 5A + 6.

Q23. If A = , find a matrix X such that AX = 2.

Q24. If A = , prove that A3 – 6A2 + 7A + 2 = 0

Q25. (i) If A = and B = show that (A + B)2 = A2 + B2.

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(ii) If A = and B = and (A + B) (A-B) = A2 – B2, then find x and y.

Q26. If A = and I = , then show that I + A = (I – A)

Type 4: To find the Inverse of a given matrix.Inverse of a matrix

Definition: If A is a square matrix of order m and if there exists another square matrix B of the same order m such that AB = BA = I, where I is the identity matrix of order m, then B is called as the inverse of A and is denoted by A-1.

Since B is denoted as A-1 , we getAA-1 = A-1 A= I

(A) Using matrix equation.Q27. If A = , show that A2 - 3A + I = 0 Hence, find A-1.

Q28. If A = , show that A2 – 5A + 7I = 0. Hence find A-1.

Q29. If A = , then show that A2 – 4A + I = 0. Hence, find A-1.

Q30. If A = , without finding A-1, show that A-1 = show that A-1= (A – 5I)

(B) To find the inverse of a non – singular matrix by Adjoint method.Definition: The adjoint of a square matrix A = is defined as the transpose of the matrix [Aij]m x m where Aij is the co-factor of the element aij of A, for all i and j where i, j = 1,2,…. m

The adjoint of the matrix A is denoted by adj A.

For example, if A is a square matrix of order 3 x 3 then the matrix of its co-factors is

and the required adjoint of A is the transpose of the above matrix. Hence

adj A =

Definition : If A = [aij]mxm is a non – singular square matrix, then its inverse exists and it is given as Where l A l is the value of the determinant of matrix A.

Q31. Find the co-factors of the elements of the following matrices.

(i) (ii)

Q32. Find the matrix of co-factors for the following matrices.

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(i) (ii)

Q33. Find the adjoint of the following matrices.

(i) A = (ii)

Q34. If A = , verify that A (adj A) = (adj A) A = l A l I

Q35. Find the inverse of each of the following matrices by the adjoint method.(i) (ii)

(iii) (iv)

Q36. Find the inverse of the following matrices by adjoint method.

(i) (ii) (iii)

(C) To find the inverse of a non – singular matrix by elementary transformation.

Q37. Find which of the following matrices are invertible.(i) (ii) (iii)

(iv) (v) (vi)

(vi) (viii) (ix)

Q38. Find AB if A = and B = . Hence, determine if AB has the inverse.

Q39. (i) If A = then reduce it to I3 by using column transformations.

(ii) If A = then reduce it to I3 by using row transformations.

Q40. If A = is a nonsingular matrix, find A-1 by using elementary row

transformations. Hence, find the inverse of

Q41. If A = and X is a 2 x 2 matrix such that AX = I, then find X

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Q42. Find the inverse of each of the following matrices (if they exist) using elementary transformations.

(i) (ii) (iii) (iv) (v)

(vi) (vii) (viiii) (ix) (x)

Type 5 : To solve give system of equations. (A) By the method of Reduction

Method of reduction, for solving a system of linear equations.From the name it is clear that, in this method the given equations are reduced to

a certain form to get the solution.For this method also we first write the given equations in the form of a matrix equation AX = B.Then we perform the suitable row transformation on the matrix A (note that we cannot use column transformations) and reduce it to an upper triangular matrix. (Refer to the types of matrices). The same transformations are performed simultaneously on the matrix B.

Solve the following equations by the method of reduction.Q43. 2x+ y = 5 3x + 5y = -3

Q44. x + 3y = 2 3x + 5y = 4

Q45. 3x – y = 1 4x + y = 6

Q46. 5x + 2y = 4 7x + 3y = 5

Q47. x – y + z = 1, 2x – y = 1 and 3x + 3y – 4z = 2

Q48. x+ y = 1, y + z = z +x =

Q49. x + y + z = 6, 3x – y + 3z = 10 and 5x + y – 4z = 3

Q50. 2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1

(B) By Method of InversionFor solving a system of liner equations, from the name of the method, it is clear that, here we are going to use the inverse of a matrix. This can be done as follows.Consider the three equations:

Using matrices this system can be considered as

=

i.e. AX = B, A is of order 3 x 3, X is of order 3 x 1, B is of order 3 x 1.24


Now, if the solution of the given equations exists, then the matrix A must be nonsingular (recall that by Cramer’s rule of determinants the

Solution of the system exists, if ≠ 0

Hence, if A is non-singular, then A-1 exists.Consider AX = B and premultiply it by A-1. We get

A-1 (AX) = A-1Bi.e. (A-1 A) = A-1 B …… (by associative law)i.e. IX= A-1B …… I is identityi.e. X = A-1B ….. I is identity

Solve the following equations by the method of inversion.Q51. x + 2y = 2, 2x + 3y = 3

Q52. x + y = 4, 2x – y = 5

Q53. 2x – y = 4, 3x + 4y = 3

Q54. 2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1

Q55. 5x - y + 4z = 5, 2x + 3y + 5z = 2 and 5x – 2y + 6z = -1

Q56. x- y + z = 9 2x + 5y + 7z = 52 & 2x + y – z = 0

Type 6: Word problemsQ57 A fruit – stall has 10 dozen mangoes, 8 dozen apples and 10 dozen bananas. Their

selling prices are Rs.200, Rs. 160 and Rs. 30 per dozen respectively. Find the total amount which will be received by selling all the fruits (use matrix algebra).

Q58. A person has Rs. 30,000 which he wants to invest in fixed deposits and savings account. The interest rates for the fixed deposits and the savings account are 7% and 5% per annum. Determine how to divde Rs.30,000 in two accounts so that he gets the annual interest as (a) Rs.1800 (b) Rs. 2000.

Q59. In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III. For boys the number of units of types I, II and III are 80, 70 and 65 respectively in factory A and 85, 65 and 72 respectively in factory B. For girls the number of units of types I, II and III are 80, 75, 90 respectively in factory A and 50, 55, 80 respectively in factory B. Express the information in terms of matrices and using matrix algebra answer the following questions.

(a) How many total units of type I are produced for boys?(b) What is the total production of each type for boys and for girls?

Q60. A manufacturer produces three products A, B, C which he sells in two markets I and II. The annual sales are as shown below.

ProductsMarket A B C

I 5000 3000 4000II 6000 10000 8000

(a) If the unit sale prices of A, B, C are Rs.3, Rs.2 and Rs.4 respectively, in market I and they are Rs. 4, Rs. 3 and Rs.6 in market II. Find the total revenue in each market using matrix algebra.

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(b) If the unit cost prices of A, B, C are Rs. 2.50, Rs.1.50 and Rs.3 Respectively then find the gross profit using matrix algebra.

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reliable.redik.inreliable.redik.in/uploads/documents/Matrices_-_2.doc · Web viewQ50. 2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1 (B) By Method of Inversion For solving