Slide 1: Matrix Operations INTRODUCTION: INTRODUCTION The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. APPLICATIONS: APPLICATIONS Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analysing the results of an experiment etc. Slide 4: Also, many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology and industrial management. MATRIX NOTATION: MATRIX NOTATION Suppose we wish to express the information that has Anita15 notebooks. We may express it as  with the understanding that the number inside [ ] is the number of notebooks that Anita has. Now, if we have to express that Anita has 15 notebooks and 6 pens. We may express it as [15 6] OR as a column matrix. with the understanding that at first number inside [ ] is the number of notebooks while the other one is the number of pens possessed by Anita . Slide 6: Let us now suppose that we wish to express the information of possession of notebooks and pens by Radha and her two friends Fauzia and Simran which is as follows: Radha has 15 notebooks and 6 pens, Fauzia has 10 notebooks and 2 pens, Simran has 13 notebooks and 5 pens. DEFINITION: DEFINITION A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters. The following are some examples of matrices: Slide 8: In the above examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2 rows and 3 columns. Slide 9: MATRIX: A rectangular arrangement of numbers in rows and columns. The ORDER of a matrix is the number of the rows and columns. The ENTRIES are the numbers in the matrix. What is a Matrix? rows columns This order of this matrix is a 2 x 3. Slide 10: What is the order? 3 x 3 3 x 5 2 x 2 4 x 1 1 x 4 (or square matrix) (Also called a row matrix) (or square matrix) (Also called a column matrix) Slide 11: To add two matrices, they must have the same order. To add, you simply add corresponding entries. Adding Two Matrices Slide 12: = = 7 7 4 5 0 7 5 7 Slide 13: Subtracting Two Matrices To subtract two matrices, they must have the same order. You simply subtract corresponding entries. Slide 14: = 5-2 -4-1 3-8 8-3 0-(-1) -7-1 1-(-4) 2-0 0-7 = 2 -5 -5 5 1 -8 5 3 -7 Slide 15: Multiplying a Matrix by a Scalar In matrix algebra, a real number is often called a SCALAR . To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar. Slide 16: -2 6 -3 3 -2(-3) -5 -2(6) -2(-5) -2(3) 6 -6 -12 10 EXAMPLES: EXAMPLES Example 1 Consider the following information regarding the number of men and women workers in three factories I, II and III Men workers Women workers I 30 25 II 25 31 III 27 26 Represent the above information in the form of a 3 × 2 matrix. What does the entry in the third row and second column represent? Example 2 If a matrix has 8 elements, what are the possible orders it can have? Example 3 Construct a 3 × 2 matrix whose elements are given by TYPES OF MATRICES: TYPES OF MATRICES Slide 21: A matrix U is an upper triangular matrix if its nonzero elements are found only in the upper triangle of the matrix, including the main diagonal; that is: uij = 0 if i > j A matrix L is an lower triangular matrix if its nonzero elements are found only in the lower triangle of the matrix, including the main diagonal; that is: lij = 0 if i < j A matrix U is an upper triangular matrix if its nonzero elements are found only in the upper triangle of the matrix, including the main diagonal; that is: uij = 0 if i > j A matrix L is an lower triangular matrix if its nonzero elements are found only in the lower triangle of the matrix, including the main diagonal; that is: lij = 0 if i < j Slide 22: A matrix U is an upper triangular matrix if its nonzero elements are found only in the upper triangle of the matrix, including the main diagonal; that is: u ij = 0 if i > j A matrix L is an lower triangular matrix if its nonzero elements are found only in the lower triangle of the matrix, including the main diagonal; that is: l ij = 0 if i < j Properties of matrix addition : Properties of matrix addition (i) Commutative Law: If A = [ aij ], B = [ bij ] are matrices of the same order, say m × n , then A + B = B + A. ( II)ASSOCIATIVE Law : For any three matrices A = [ aij ], B = [ bij ], C = [ cij ] of the same order, say m × n , (A + B) + C = A + (B + C). (III) Existence of additive identity: Let A = [ aij ] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition. (iv) The existence of additive inverse: Let A = [ aij ] m × n be any matrix, then we have another matrix as – A = [– aij ] m × n such that A + (– A) = (– A) + A= O. So – A is the additive inverse of A or negative of A. Properties of scalar multiplication of a matrix : Properties of scalar multiplication of a matrix If A = [ a ij ] and B = [ b ij ] be two matrices of the same order, say m × n , and k and l are scalars, then (i) k (A +B) = k A + k B, (ii) ( k + l )A = k A + l A Hint: : Suppose Meera and Sara are two friends. Meera wants to buy 2 pens and 5 story books, while Sara needs 8 pens and 10 story books. They both go to a shop to enquire about the rates which are quoted as follows: Pen – Rs 5 each, story book – Rs 50 each. How much money does each need to spend? Clearly, Meera needs Rs (5 × 2 + 50 × 5) that is Rs 260, while Sara needs (8 × 5 + 50 × 10) Rs, that is Rs 540 How will you represent this in matrix form? Introduction Introduction for matrix multiplication Slide 26: Suppose that they enquire about the rates from another shop, quoted as follows: pen – Rs 4 each, story book – Rs 40 each. Now, the money required by Meera and Sara to make purchases will be respectively Rs (4 × 2 + 40 × 5) = Rs 208 and Rs (8 × 4 + 10 × 40) = Rs 432 How will you represent this information in matrix form? Matrix Multiplication: Matrix Multiplication For multiplication of two matrices A and B, the number of columns in A should be equal to the number of rows in B. Furthermore for getting the elements of the product matrix, we take rows of A and columns of B, multiply them element-wise and take the sum. Slide 28: Example EXAMPLE Remark If AB is defined, then BA need not be defined. In the above example, AB is defined but BA is not defined because B has 3 column while A has only 2 (and not 3) rows. If A, B are, respectively m × n , k × l matrices, then both AB and BA are defined if and only if n = k and l = m . In particular, if both A and B are square matrices of the same order, then both AB and BA are defined. Non-commutativity of multiplication of matrices : Non-commutativity of multiplication of matrices REMARK: REMARK In the above example both AB and BA are of different order and so AB ≠ BA. But one may think that perhaps AB and BA could be the same if they were of the same order. But it is not so, here we give an example to show that even if AB and BA are of same order they may not be same. EXAMPLE: EXAMPLE Slide 32: Zero matrix as the product of two non zero matrices We know that, for real numbers a , b if a X b = 0, then either a = 0 or b = 0. This need not be true for matrices , we will observe this through an example . Example Properties of multiplication of matrices : Properties of multiplication of matrices The associative law: For any three matrices A, B and C. We have (AB) C = A (BC), whenever both sides of the equality are defined. 2. The distributive law: For three matrices A, B and C. (i) A (B+C) = AB + AC (ii) (A+B) C = AC + BC, whenever both sides of equality are defined. 3. The existence of multiplicative identity: For every square matrix A, there exist an identity matrix of same order such that IA = AI = A. Slide 34: Transpose of a Matrix Slide 35: Properties of transpose of the matrices Slide 36: Symmetric and Skew Symmetric Matrices Slide 39: Example : Express the matrix B = as the sum of a symmetric and a skew symmetric matrix. Slide 41: Elementary Operation (Transformation) of a Matrix There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations . (i) The interchange of any two rows or two columns . Symbolically the interchange of i th and j th rows is denoted by R i ↔ R j and interchange of i th and j th column is denoted by C i ↔ C j . Slide 42: (ii)The multiplication of the elements of any row or column by a non zero number . Symbolically, the multiplication of each element of the “ i” th row by k ,where k ≠ 0 is denoted by R i → k R i . The corresponding column operation is denoted by C i → k C i (iii)The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Symbolically, the addition to the elements of i th row, the corresponding elements of j th row multiplied by k is denoted by R i → R i + k R j . The corresponding column operation is denoted by C i → C i + k C j . Invertible Matrices : Invertible Matrices 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, then B is called the inverse matrix of A and it is denoted by A – 1 . In that case A is said to be invertible. NOTE: A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order. If B is the inverse of A, then A is also the inverse of B Slide 44: Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. Proof : Let A = [ aij ] be a square matrix of order m . If possible, let B and C be two inverses of A. We shall show that B = C. Since B is the inverse of A AB = BA = I ... (1) Since C is also the inverse of A AC = CA = I ... (2) Thus B = BI = B (AC) = (BA) C = IC = C Slide 46: Inverse of a matrix by elementary operations (i)Let X, A and B be matrices of, the same order such that X = AB. In order to apply a sequence of elementary row operations on the matrix equation X = AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS. (ii)Similarly, in order to apply a sequence of elementary column operations on the matrix equation X = AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS. (iii)To find A –1 using elementary row operations, write A = IA and apply a sequence of row operation on A = IA till we get, I = BA. The matrix B will be the inverse of A. Similarly, if we wish to find A –1 using column operations, then, write A = AI and apply a sequence of column operations on A = AI till we get, I = AB. REMARK: REMARK In case, after applying one or more elementary row (column) operations on A = IA (A = AI), if we obtain all zeros in one or more rows of the matrix A on L.H.S.,then A –1 does not exist. Slide 48: Algorithm for finding the inverse of a Matrix by Elementary Transformations Obtain the square matrix , say A whose order is 3x 3 Write A = I 3 A Introduce 1 at a 11 either by interchanging two rows or adding a constant multiple of elements of some other row to first row. After introducing 1 at a 11 introduce zeroes at all other places in first column. Introduce 1 at a 22 with the help of 2 nd and 3 rd row. After introducing 1 at a 22 introduce zeroes at all other places in 2nd column. 7) Introduce 1 at a 33 with the help of 3rd row and 3rd column. 8) After introducing 1 at a 33 introduce zeroes at all other places in third column.