CU-BCA-SEM-I-Mathematics

2.2 MULTIPLICATION OF MATRICES Multiplication of Matrices: Let A and B be two matrices. If the number of columns in matrix A is equal to the number of rows in matrix B is called their product AB. 2.3 PROPERTIES OF MULTIPLICATION OF MATRICES Properties of Multiplication of Matrices: 1. Commutative Law Generally, AB ≠ BA 2. Associative Law (AB)C = A(BC) 51 CU IDOL SELF LEARNING MATERIAL (SLM)

3. Existence of multiplicative Identity AI = A = IA where I is called multiplicative Identity matrix. 4. Distributive Law A(B + C) = AB + AC 5. Cancellation Law Let us consider A is non-singular matrix then AB = AC ⇒ B = C (left cancellation law) BA = CA ⇒B = C (right cancellation law) 6. AB = 0, does not necessarily imply that A = 0 or B = 0 or both A and B = 0 Important Points to be Remembered (i) If A and B are square matrices of the same order in n then both the product AB and BA are defined, and each is a square matrix of order n. (ii) If the matrix product AB, then the matrix A is called pre multiplier and B is called post multiplier. (iii) Generally, the rules of multiplication of matrices is row column wise (or → ↓ wise) the first row of AB is obtained by multiplying the first row of A with first, second, third, columns of B respectively; similarly second row of A with first, second, third, … columns of B, respectively and so on. 52 CU IDOL SELF LEARNING MATERIAL (SLM)

2.4 SQUARE MATRIX Definition: Square matrix is a matrix that has an equal number of rows and columns. In mathematics, m × m matrix is called the square matrix of order m. If we multiply or add any two square matrices, the order of the resulting matrix remains the same. Example:  2  9 6 3 5 4 Let A =  2 0 4 is square matrix of order 3. Properties of Multiplication Positive Integral Powers of a Square Matrix Let A be a square matrix. Then, we can define i. An + 1 = An. A, where n ∈ N. ii. Am. An = Am + n iii. (Am)n = Amn, ∀ m, n ∈ N 2.5 EXAMPLE IN MULTIPLICATION OF MATRICES Example: 1 53 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 54 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 2 Solution: 56 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 3 59 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example: 4 60 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 5 62 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 6 Solution: 65 CU IDOL SELF LEARNING MATERIAL (SLM)

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2.6 SUMMARY ● The identity matrix of every square matrix is a diagonal stretch of 1s from the upper- left corner to the lower-right corner, with all other members set to 0. ● Multiplying matrices is possible when inner dimensions are the same and the number of columns in the first matrix must match the number of rows in the second. ● There is no identity for non-square matrices. That is, there is no matrix such as [A][I]=[I][A]=[A] given a non-square matrix [A]. ● It is also feasible to multiply two matrices together; however, matrices can only be multiplied if the first matrix's number of columns equals the second matrix's number of rows. The two matrices can be multiplied if they meet this requirement. The outcome is a third matrix with the same number of rows as the first matrix and the same number of columns as the second matrix. 2.7 KEYWORD • Commutative : involving the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved, e.g. a × b = b × a. • Associative: involving the condition that a group of quantities connected by operators gives the same result whatever their grouping e.g. ( a × b ) × c = a × ( b × c ). 67 CU IDOL SELF LEARNING MATERIAL (SLM)

• Identity : an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. • Inverse: a function or operation which reverses the order or operation of another function or operation”. • Square matrix : is a matrix with the same number of rows and columns. 2.8 LEARNING ACTIVITY  1 1  1 2 13 B  2 1 A  2 1   2 3 , write down the matrix AB. Would it be 1. If and possible to find the product of BA? If so, compute it, and if not, give reasons. ___________________________________________________________________________ _____________________________________________________________________ A   1 2  B   2 12 . 2 1 1 2. Find AB if and ___________________________________________________________________________ _____________________________________________________________________ A   3 01 B   1 2  2 3 0 3. Check whether AB = BA if the matrices and . 68 CU IDOL SELF LEARNING MATERIAL (SLM)

___________________________________________________________________________ _____________________________________________________________________  1 2 4  2 1 0 4. Find A2 from the matrix  5 4 1 ___________________________________________________________________________ _____________________________________________________________________ 5. Multiply the matrix  3 8  1 2  . 0 4 4 1 ___________________________________________________________________________ _____________________________________________________________________ 2.9 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. Multiply  8 91 2 03 . 5 4  3 13 x    14 6 y 2. Find the value of x and y 69 CU IDOL SELF LEARNING MATERIAL (SLM)

  2 1 7  and B  4 1 5 A 3 1 0 3. If possible, find BA and AB  0 2 1 A   7 52, B  02 13 . 1 4. Find AB if  3 1  0 1 2  B  1 3 A  4 11 2  6 1  find AB. 5. If and Long Questions  6 6 7 6 2 5 1. Find A2 from the matrix 7 5 1 .  0 2  2  6 6 7  A   2 5  2 B   6 2  5 2. Find AB if  2 2 0  and   7 5 1  .  2 1 A  3 0   4 5 23 3. Find AB if  5 2  and B  1 0 70 CU IDOL SELF LEARNING MATERIAL (SLM)

 0 6  2 A   2 2  2   2  2 0  . 4. Find A2 if Find BA if A  85 76 and B   0 1  4 2 5. B. Multiple Choice Questions 1. For any square matrix A, AAT is a a. unit matrix b.symmetric matrix c.skew-symmetric matrix d. Diagonal matrix 2. If A is any square matrix, then which of the following is skew-symmetric? a.A + AT b.A – AT c.AAT d.ATA 71 CU IDOL SELF LEARNING MATERIAL (SLM)

3. If A  13 34 and A2 – KA – 5I = 0, then K = a.5 b.3 c.7 d.8  a b c  A  b c a  c a b  where a, b, c are real positive numbers, abc = 1 and ATA = 4. If matrix I, then the value of a3 + b3 + c3 is a. 1 b. 2 c. 3 d.4 A   cos x  sin x  sin x cos x 5. If , find AAT. a. Zero Matrix b. I2 72 CU IDOL SELF LEARNING MATERIAL (SLM)

c. 11 11 d.Nоne of these Answers 1-b, 2-b, 3-a. 4-d, 5-b 2.10 REFERENCES References book ● Vittal, P.R, “Allied Mathematics”, Reprint,Margham Publications, Chennai. ● Venkata chalapathy, S.G, “Allied Mathematics”, Margham Publications, Chennai. Textbook references ● Singaravelu, A. “Allied Mathematics”, Meenakshi Agency, Chennai. ● N. Herstein, Topics in Algebra, John Wiley and Sons, 2015. ● Gilbert Strang, Introduction to linear algebra, Fifth Edition, ANE Books, 2016. 73 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT - 3: MATRIX3 STRUCTURE 3.0 Learning Objectives 3.1 Introduction 3.2 Rank of a matrix 3.2.1 Properties of Rank of matrix 3.3 Transpose of a matrix 3.4 Worked example 3.5 Summary 3.6 Keywords 3.7 Learning Activity 3.8 Unit End Questions 3.9 References 3.0 LEARNING OBJECTIVES After studying this unit, you will be able to: ● Recognize how the order of a matrix changes when its transpose is discovered. ● Explain how to transpose matrices of different order. ● Identify symmetric and skew-symmetric matrices. 74 CU IDOL SELF LEARNING MATERIAL (SLM)

● Calculate transpose matrices and perform matrix calculations. 3.1 INTRODUCTION In this section, discuss about the systems of linear equations we mentioned that a system can have no solutions, a unique solution, or infinitely many solutions. In this section we’re going to introduce an invariant1 of matrices, and when this invariant is computed for the matrix of coefficients of a system of linear equations, it can give us information about the number of solutions to this system. We should have got knowledge of sub-matrices and minors of a matrix. Consider A be a given matrix. Matrix obtained by deleting some rows and some columns or matrix A is known as the sub-matrix of A. A matrix is called a sub-matrix of itself as it is obtained by leaving zero number of rows and columns. Minor of the matrix is the determinant of the square matrix that is obtained by deleting one row and one column from some larger matrix. In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal. i.e., it switches the row and column indices of the matrix A by producing another matrix and it is denoted by AT . In the year 1858, the transpose of a matrix was introduced by the British mathematician Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT. 3.2 RANK OF MATRIX Rank of a Matrix: 75 CU IDOL SELF LEARNING MATERIAL (SLM)

The maximum of number its linearly independent columns or rows of a matrix is called the rank of a matrix. The rank of a matrix cannot exceed the number of its rows or columns. Let us consider a square matrix, the columns (rows) are linearly independent only if the matrix is nonsingular. A matrix is said to be of rank zero when all of its elements become zero.The rank of the null matrix is zero.A positive integer r is said to be the rank of a non-zero matrix A, if (a) There exists at least one minor in A of order r which is not zero. (b) Every minor in A of order greater than r is zero, rank of a matrix A is denoted by ρ(A) = r. 3.2.1 Properties of Rank of a Matrix: Properties of Rank of a Matrix: 1. The rank of a null matrix is zero ie, ρ(0) = 0 2. If In is an identity matrix of order n, then ρ(In) = n. 3. (a) If a matrix A doesn’t possess any minor of order r, then ρ(A) ≥ r. (b) If at least one minor of order r of the matrix is not equal to zero, then ρ(A) ≤ r. 4. If every (r + 1)th order minor of A is zero, then any higher order – minor will also be zero. 5. If A is of order n, then for a non-singular matrix A, ρ(A) = n 6. ρ(A’)= ρ(A) 76 CU IDOL SELF LEARNING MATERIAL (SLM)

7. ρ(A*) = ρ(A) 8. ρ(A + B) = ρ(A) + ρ(B) 9. If A and B are two matrices such that the product AB is defined, then rank (AB) cannot exceed the rank of the either matrix. 10. If A and B are square matrix of same order and ρ(A) = ρ(B) = n, then p(AB)= n 11. Every skew-symmetric matrix,of odd order has rank less than its order. 12. Elementary operations do not change the rank of a matrix. Note: Echelon Form of a Matrix A non-zero matrix A is said to be in Echelon form, if A satisfies the following conditions 1. All the non-zero rows of A, if any precede the zero rows. 2. The number of zeros preceding the first non-zero element in a row is less than the number of such zeros in the successive row. 3. The first non-zero element in a row is unity. 4. The number of non-zero rows of a matrix given in the Echelon form is its rank. Example 1: Solution: 77 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 2: Solution: 78 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 3: Solution: 79 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 4: Solution: 80 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 5: Solution: 81 CU IDOL SELF LEARNING MATERIAL (SLM)

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3.3 TRANSPOSE OF A MATRIX Transpose of a Matrix Let A = [aij]m x n, be a matrix of order m x n. A matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A’ or AT. A’ = AT = [aij]n x m Example: 1 2 A  3 5  1 3 76  Let 6 7  then transpose of matrix (AT) is 2 5 Properties of Transpose 1. (A’)’ = A 2. (A + B)’ = A’ + B’ 3. (AB)’ = B’A’ 4. (KA)’ = kA’ 5. (AN)’ = (A’)N 6. (ABC)’ = C’ B’ A’ Transpose Conjugate of a Matrix 83 CU IDOL SELF LEARNING MATERIAL (SLM)

A matrix A is called transpose conjugate then the transpose of the conjugate of a matrix A and it is denoted by A0 or A*. i.e., (A’) = A’ = A0 or A* Properties of Transpose Conjugate of a Matrix (i) (A*)* = A (ii) (A + B)* = A* + B* (iii) (kA)* = kA* (iv) (AB)* = B*A* (v) (An)* = (A*)n 3.4 WORKED EXAMPLES Example: 1 Solution: 84 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 2 87 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 3 Solution: 90 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 4 91 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 5 93 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 6 95 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example: 7 96 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 8 Solution: 98 CU IDOL SELF LEARNING MATERIAL (SLM)

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3.5 SUMMARY ● The maximum number of linearly independent column vectors or row vectors in a matrix is called its rank. ● Only if the matrix is non-singular can the columns (rows) of a square matrix be demonstrated to be linearly independent. In other words, any non-singular matrix of order n has a rank of n. ● The transposed matrix's row and column dimensions are the inverse of the original matrix's dimensions. ● The product of a square matrix multiplied by a column matrix comes naturally in linear algebra for solving linear equations, and represents linear transformations. ● A rectangular arrangement of numbers or phrases that can be used for a variety of purposes, including changing coordinates in geometry. 100 CU IDOL SELF LEARNING MATERIAL (SLM)