Matrix and Determinant PDF

MATRIX AND DETERMINENT MAREICES let us start with a simple example. There are three brothers, Ajit (A), Bhola (B) and Chanchal (C) ina family. A has a set of 3 pants, 3 shirts, 2 shorts and 1 tie. B has a set of 5 pants, 5 shirts, No short...

MATRIX AND DETERMINENT MAREICES let us start with a simple example. There are three brothers, Ajit (A), Bhola (B) and Chanchal (C) ina family. A has a set of 3 pants, 3 shirts, 2 shorts and 1 tie. B has a set of 5 pants, 5 shirts, No short, and 2 tie. C has a set of 6 pants, 8 shirts, 5 shorts and no tie. we can arrange these data in the following convenient system:  MATRIX AND DETERMINENT MAREICES pant shirt short tie P S Sh T A 3 3 2 1 1st Row B 5 5 0 2 2nd Row C 6 8 5 0 3rd Row 3rd 1st 2nd 4th column column column column  MATRIX AND DETERMINENT MAREICES This system comprises of 3 rows, and 4 columns (row is written first followed by column, thus it is 3 x 4 matrix). We have written belongings of A in the first row , belongings of B in the second row, belongings of C in the third row. The first column gives the number of pants that (A),(B),(C) together have, the second column gives the total number of shirts, the third column gives the total number of shorts and the fourth column gives the total number of ties. The numbers written in such a form of rows and columns and enclosed by [ ] or ( ) (large parenthesis/ bracket) or || || (double bars) is called Matrix  MATRIX AND DETERMINENT MAREICES A matrix is defined as rectangular array of elements arranged in a row and column. General form of matrix is : A= a11 a12 ……. a1n a 21 a22 ……. a2n ………………………….. am1 am2 …… amn  MATRIX AND DETERMINENT MAREICES This Matrix [A] has (m) rows and (n) columns, it is read as (m x n) matrix 9or m by n matrix). Each element of the matrix is denoted by (aij), the first element (i) denotes row and the second element (j) denotes column. Thus, if we say a13 , it means, the element belongs to the first row and third column. Again a34 means, the element belongs to third row and fourth column. Thus aij means, the element belongs to ith row and jth column.  MATRIX AND DETERMINENT MAREICES Order of a Matrix: If a matrix has (mn) number of total elements arranged in (m) rows and (n) columns, the order of the Matrix is (m x n) ( read as m by n). Few points about the Matrix is to be remembered: 1. A matrix is not just an arrangement of numbers. In a given Matrix each element has its assigned position in a particular row and column. Thus: 1 2 3 1 2 3 2 3 1 is not same as 2 3 2 3 2 1 3 1 1 In a Matrix each element has an assigned position.  MATRIX AND DETERMINENT MAREICES Order of a Matrix: 2. Matrix [A] = matrix [B] if they have same order and each element of [A] is equal to the corresponding element of [B] That is , if 𝑎11 𝑎12 [A] = and [B] = 𝑏11 𝑏12 𝑎21 𝑎22 𝑏21 𝑏22 then A = B if and only if, a11 = b11; a12 = b12; a21 = b21 and a22 = b22 That is aij = bij  MATRIX AND DETERMINENT MAREICES Order of a Matrix: 3. A matrix of (m) row and (n) column is m x n matrix 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋮ it is (m x n) matrix 𝑎𝑚1 ⋯ 𝑎𝑚𝑛 4. If in a Matrix m=n, that is number of rows is equal to number of columns, then it is a square matrix of order (n) 1 2 3 𝑎 𝑏 A= 𝑐 𝑑 𝐵= 2 3 4 8 7 6  MATRIX AND DETERMINENT MAREICES Order of a Matrix: 5. If a matrix consist of only one column , it is a column matrix or column vector. 𝑎1 𝑎2 𝑎3 6. If in a Matrix consist of only one row , it is called a row matrix or row vector. [ a1 b1 c1]  MATRIX AND DETERMINENT MAREICES Order of a Matrix: 7. A matrix is said to be a zero or null matrix if and only if each of its elements is zero. 2 0 0 1 0 A= 0 4 𝐵= 0 4 0 0 0 7 8. Diagonal Matrix: In a square matrix, if all the diagonal elements are non- zero and the rest are zero , it is called a diagonal matrix. 1 0 2 0 0 A= B= 0 4 0 0 6 0 0 7  MATRIX AND DETERMINENT MAREICES 9. Scalar Matrix: In a square Matrix if all the diagonal elements are the same and the rest of the elements are zero, it is called scalar matrix. 2 0 0 4 0 A= 0 4 𝐵= 0 2 0 0 0 2 10. Non zero Matrix: In a matrix, if at least one element is non zero, it is called non-zero matrix. A= 1 2 B= 0 0 3 6 0 2  MATRIX AND DETERMINENT MAREICES 11. Identity Matrix: In a scalar Matrix if all the diagonal elements are one, it is called scalar matrix. 1 0 0 1 0 A= 0 1 𝐵= 0 1 0 0 0 1 12. Comparable Matrix: Two matrices are said to be comparable, if their orders are same. A= 1 2 B= 𝑎 𝑏 3 6 𝑐 𝑑 since, both the matrices are 2 x 2 matrix , they are comparable.  MATRIX AND DETERMINENT MAREICES Addition and Subtraction of matrix: If the matrix [A] and [B] are of the same order (m x n), they can be added or subtracted. If A = [aij] and B = [bij] then A + B = [aij] + [bij] = [aij + bij] A – B = [aij – bij] It means corresponding elements are added (or subtracted). The new matrix will be of same order (m x n)  MATRIX AND DETERMINENT MAREICES Addition and Subtraction of matrix: if A = 𝑎11 𝑎12 and B = 𝑏11 𝑏12 𝑎21 𝑎22 𝑏21 𝑏22 𝑎11 + 𝑏11 𝑎12 + 𝑏12 Then, A + B = 𝑎21 + 𝑏21 𝑎22 + 𝑏22 and A – B = 𝑎11 − 𝑏11 𝑎12 − 𝑏12 𝑎21 − 𝑏21 𝑎22 − 𝑏22  MATRIX AND DETERMINENT MAREICES Addition and Subtraction of matrix: if A = 1 3 and B = 2 3 6 5 4 1 1+2 3+3 3 6 Then, A + B = 6+4 5+1 = 10 6 and A – B = 1 − 2 3 − 3 = −1 0 6−4 5−1 2 4  MATRIX AND DETERMINENT MAREICES Properties of matrix Addition: 1. Matrix addition is commutative. A+B=B+A 2. Matrix addition is associative (A + B ) + C = A + ( B + C) when C is the third matrix. 3. A + 0 = 0 + A = A Thus, zero/null matrix plays the same role in matrix algebra as zero in ordinary algebra.  MATRIX AND DETERMINENT MAREICES Properties of matrix Addition: by extending this definition of addition we can define scalar multiplication as: 𝑎11 𝑎12 ⋯ 𝑎1𝑛 𝑎21 …. 𝑎22 …. … 𝑎2𝑛 …. If A = ….. …. …. 𝑎𝑚1 𝑎𝑚2 𝑎𝑚𝑛 Then A + A +A + ….. + A k-times 𝑎11 𝑎12 ⋯ 𝑎1𝑛 𝑎21 …. 𝑎22 …. … 𝑎2𝑛 …. = kA = k ….. …. …. 𝑎𝑚1 𝑎𝑚2 𝑎𝑚𝑛  MATRIX AND DETERMINENT MAREICES Properties of matrix Addition: 𝑎11 𝑎12 ⋯ 𝑎1𝑛 𝑎21 …. 𝑎22 …. … 𝑎2𝑛 …. kA = k ….. …. …. 𝑎𝑚1 𝑎𝑚2 𝑎𝑚𝑛 𝑘𝑎11 𝑘𝑎12 ⋯ 𝑘𝑎1𝑛 𝑘𝑎21 …. 𝑘𝑎22 …. … 𝑘𝑎2𝑛 …. = ….. …. …. 𝑘𝑎𝑚1 𝑘𝑎𝑚2 𝑘𝑎𝑚𝑛 That is each element of the matrix is multiplied by (k)  MATRIX AND DETERMINENT MAREICES Properties of matrix Addition: The following results can also be established: 1. A + (-A) = (-A) + A = 0 2. A + A = 2A 3. K(A + B ) = kA + kB 4. (k1 + k2) A = K1A + k2A  MATRIX AND DETERMINENT MAREICES Problems on matrix:  MATRIX AND DETERMINENT MAREICES Matrix Multiplication: If the number of columns of the first matrix is equal to the number of rows of the second matrix, we can find out the ptoduct, otherwise it is not defined. if A = [aij](mxn) and B = [bij](nxp) Then AB = [Cij](mxp) Where Cij is the product of ith row of matrix A and jth column of matrix B 𝑎11 𝑎12 𝑏11 𝑏12 Thus, A = 𝑎21 𝑎22 and B = 𝑏21 𝑏22  MATRIX AND DETERMINENT MAREICES Matrix Multiplication: 𝑎11 𝑎12 𝑏11 𝑏12 Thus, A = 𝑎21 𝑎22 and B = 𝑏21 𝑏22 𝑎11𝑏11 + 𝑎12𝑏21 𝑎11𝑏12 + 𝑎12𝑏22 Then AB = 𝑎21𝑏11 + 𝑎22𝑏21 𝑎21𝑏12 + 𝑎22𝑏22  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: Suppose the three brothers [Ajit (A), Bhola (B) and Chanchal (C)] are interested to purchase some more cloth. They observed that the prices of different items in a shop (S1) is as follows: 1. Per Pant Rs 50.00 2. Per Shirt Rs 30.00 3. Per Short Rs 25.00 4. Per Tie Rs 20.00  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: Let us write the prices of these items in the form of a column matrix. 50 Price per pant 30 Price per shirt 25 Price per short 20 Price per tie Say, (A) wishes to purchase 2 pants (P), 2 shirts (S), 1 short (Sh) and 2 ties (T) at that price from the shop. How much will he pay for the items? Put the quantities that (A) wants to purchase as a row [ 2 2 1 2]  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: Say, (A) wishes to purchase 2 pants (P), 2 shirts (S), 1 short (Sh) and 2 ties (T) at that price from the shop. How much will he pay for the items? Put the quantities that (A) wants to purchase as a row [ 2 2 1 2] Shorts (Sh) Pant (P) Shirt (S) Tie (T) The amount Ajit (A) has to pay for the items can be calculated by multiplying the quantities of pant(P), Shirt (S), shorts (Sh) and tie(T) with the price and then adding them up.  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: Ajit (A) will pay = [ 2 2 1 2] x 50 Pant (P) Shorts Pant (P) (Sh) 30 Shirt (S) Shirt (S) Tie (T) 25 Shorts (Sh) 20 Tie (T) This can be written as: (2 x 50) + (2 x 30) + ( 1 x 25) + ( 2 x 20) = Similarly , say Bhola (B) wants to purchase [1 2 2 1] Shorts Pant (P) (Sh) Shirt (S) Tie (T)  MATRIX AND DETERMINENT MAREICES Shorts (Sh) Multiplication of a Matrix by a Column Matrix: Tie (T) Shirt (S) [1 2 2 1] x 50 = (1 x 50) + (2 x 30) + (2 x 25) + ( 1 x 20) Pant (P) 30 = 25 20 Say, Chanchal (C) wants to purchase [3 3 0 5], at the same price; then (C) has to pay  MATRIX AND DETERMINENT MAREICES Shorts (Sh) Multiplication of a Matrix by a Column Matrix: Tie (T) Shirt (S) [3 3 0 5] x 50 = (3 x 50) + (3 x 30) + (0 x 25) + ( 5 x 20) Pant (P) 30 = 25 20 The whole thing can be written as:  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: 2 2 1 2 1 2 2 1 X 50 Price per Pant (P) 3 3 0 5 Price per Shirt (S) Pant (P) Shorts (Sh) 30 Price per Shorts (Sh) Shirt (S) Tie (T) 25 Price perTie (T) 20 The whole thing can be written as:  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: 2 2 1 2 1 2 2 1 X 50 = (2 x50)+(2 x 30)+(1 x 25)+(2 x 20) 3 3 0 5 30 (1 x 50)+(2 x 30)+(2 X 25)+(1 x 20) 25 (3 x 50)+(3 x 30)+(0 x 25)+(5 X 20) 20 = 225 180 340  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by a Column Matrix: Using symbols in place of numerical, we have 𝑎11 𝑎12 𝑎13 𝑎14 b1 𝑎21 𝑎22 𝑎23 𝑎24 x b2 𝑎31 𝑎32 𝑎33 𝑎34 b3 b4 𝑎11𝑏1 + 𝑎12𝑏2 + 𝑎13𝑏3 + 𝑎14𝑏4 Or AB = 𝑎21𝑏1 + 𝑎22𝑏2 + 𝑎23𝑏3 + 𝑎24𝑏4 𝑎31𝑏1 + 𝑎32𝑏2 + 𝑎33𝑏3 + 𝑎34𝑏4 The elements of first row of matrix AB is obtained by multiplying elements of first row of matrix A by corresponding element of matrix B and adding. For this, it is necessary that the number of elements of each row in Matrix A must be equal to number of elements in columns of Matrix B.  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: Noe, let us consider a slightly more difficult situation. Suppose the three brothers (Ajit (A), Bhola (B), Chanchal (C)) find different prices of pant (P), Shirt (S), Shorts (Sh) and Tie (T) at different shops S1, S2, S3 and S4 as given: S1 S2 S3 S4 P 50 45 40 35 s 30 35 30 40 Sh 25 20 20 30 T 20 25 2 20 We assume that that the three brothers (A, B, and C) wants to spend minimum amount on whole lot of clothes (Pant, Shirt, Shorts and Tie). Our question is from which shop A, B and C should purchase their respective lots, assuming that the whole lot is to be purchased from the same shop.  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: A has four alternatives; he can purchase the lot [2 2 1 2] from shop S1, S2, S3 or S4 If A purchase from shop S1: ‘A’ would pay: 50 [2 2 1 2] x 30 = (2 x 50 ) + (2 x 30) + (1 x 25) +(2 x 20) = 25  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: If A purchase from shop S2: ‘A’ would pay: 45 [2 2 1 2] x 35 = (2 x 45 ) + (2 x 35) + (1 x 20) +(2 x 25) = 20 25 If A purchase from shop S3: 40 [2 2 1 2] x 30 = (2 x 40 ) + (2 x 30) + (1 x 30) +(2 x 20) = 30 20  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: If A purchase from shop S4: ‘A’ would pay: 35 [2 2 1 2] x 40 = (2 x 35 ) + (2 x 40) + (1 x 30) +(2 x 30) = 30 30 Therefore, ‘A’ would purchase from shop S3, because at this shop he pays the minimum amount (Rs 210.00)  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: In the same way we can find out the total amount spend by ‘B’ and ‘C’ for their respective lots at different shops (S1, S2, S3 and S4) by multiplying the two matrices. A 2 2 1 2 50 45 40 35 P 1 2 2 1 and 30 35 30 40 S B 25 20 30 30 Sh C 3 5 0 5 20 25 20 30 T P S Sh T S1 S2 S3 S4  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: 2 2 1 2 50 45 40 35 1 2 2 1 x 30 35 30 40 = 25 20 30 30 3 5 0 5 20 25 20 30 (2x50)+ (2x30)+(1x25)+(2x20) (2x45)+(2x35)+(1x20)+(2x25) (2x40)+(2x30)+(1x30)+(2x20) (2x35)+(2x40)+(1x30)+(2x30) (1x50)+(2x30)+(2x25)+(1x20) (1x45)+(2x35)+(2x20)+(1x25) (1x40)+(2x30)+(2x30)+(1x20) (1x35)+(2x40)+(2x30)+(1x30) (3x50)+ (5x30)+(0x25)+(5x20) (3x45)+(5x35)+(0x20)+(5x25) (3x40)+(5x30)+(0x30)+(5x20) (3x35)+(5x40)+(0x30)+(5x30) 225 230 210 240 A = 180 180 180 205 B 340 365 310 375 C S1 S2 S3 S4  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: That is, the Row1 gives the different amount that ‘A’ will pay for the same lot at four different shops. Row2 is for ‘B’ and Row3 is for ‘C’. So, ‘A’ and ‘C’ would prefer to purchase from Shop3, ‘B’ can purchase from S1, S2 or S3, since at the three shops ‘C’ pays the same amount. This example helps us to define Matrix Multiplication. Matrix multiplication is some what different from ordinary algebraic multiplication. * Firstly all the matrix can not be multiplied by each other. * Matrix multiplication is generally not commutative, that is, AB ≠ BA * Matrix A and matrix B can be multiplied if the two matrix are conformable.  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: What conformable means; it means if matrix A has order (mxn) (means (m) rows and (n) columns) and order of matrix B is (pxq), then order pair AB of the matrix A and B is said to be conformable if n = p. That is the number of column of matrix A is equal to number of rows of matrix B 1 3 2 1 2 A= 2 4 3 and B = 4 5 3 5 0 2 6 Matrix A has three columns and matrix B has three rows, hence conformable. If n ≠ p, then Matrix A and matrix B are not conformable in order A and B and we can not multiply Matrix A and B.  MATRIX AND DETERMINENT MAREICES Multiplication of a Matrix by another Matrix: * What about BA? Is it defined? It is not defined, because the number of column of B (q=2) is not equal to number of rows of Matrix A (m=3) 𝑎11 𝑎12 𝑎13 A= 𝑎21 𝑎22 𝑎23 it is 3x3 matrix 𝑎31 𝑎32 𝑎33 𝑏11 𝑏12 B= 𝑏21 𝑏22 it is 3x2 matrix 𝑏31 𝑏32 Hence, AxB will be 3x2 matrix (m=3 and q=2)