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Matrix & Deterninants Notes

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Published in: Mathematics
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This note contains all important concept and formula of Matrix and Determinants.

Pawan K / Delhi

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Qualification: B.Tech/B.E. (RGTU - 2008)

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  1. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] Important Terms , Definitions & Formulae 01. Matrix — a basic introduction: A matrix having m rows and n columns is called a matrix of order m >< n(read as" m by n"matrix).And a matrix A of order m x n is depicted as A Also in general , aij means an element lying in the ith row and J th column. No. of elements in the matrixA ]mxn is given as (m)(n) . 02.Types of Matrices: a) Column matrix : A matrix having only one column is called a column matrix or column vector. 0 General notation: A e.g. 1 Y m xl -2 b) Row matrix : A matrix having only one row is called a row matrix or row vector. e.g. [-1 2 4]1x4 , [2 5 0]1X3 General notation: A LJ Ixn c) Square matrix: It is a matrix in which the number of rows is equal to the number of columns i.e. , an m >< n matrix is said to constitute a square matrix if m = n and is known as a square matrix of order 'n' . is a square matrix of order 3. General notation: A e.g. 0 6 2 ij nxn ]mxm d) Diagonal matrix : A square matrix A is said to be a diagonal matrix if aiJ i.e. , all its non-diagonal elements are zero. 100 is a diagonal matrix of order 3. e.g. 0 6 0 005 Note:l. Also there is one more notation specifically used for the diagonal matrices. For instance consider the matrix depicted above,it can be also written as diag (1 6 5) . of order m are said to 2. The elements all, an, a33, .... , amm of a square matrix A = a• constitute the principal diagonal or simply the diagonal of the square matrix A. And these elements are known as diagonal elements of matrix A. e) Scalar matrix: A diagonal matrix A is said to be a scalar matrix if its diagonal elements are when i k, when i = j for some constant k 500 is a scalar matrix of order 3. e.g. 0 5 0 005 f) Unit or Identity matrix : A square matrix in which elements in the diagonal are all 1 and rest are all zero is equal i.e. , aiJ - called an identity matrix. is said to be an identity matrix if aiJ In other word the square matrix A = a• Y m Xm Page 1 of 7 1, if i = j
  2. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] The identity matrix of order m denoted by Imor I . 100 e.g. 13 010 001 g) Zero matrix or Null matrix: A matrix is said to be a zero matrix or null matrix if each of its elements is 02 = zero . It is denoted by Z oro . 000 e.g. 03 000 000 , O = [O]IXI h) Horizontal matrix: Am >< n matrix is said to be a horizontal matrix if m < n . 42 l) Vertical matrix: A m >< n matrix is said to be a vector matrix if m > n. e.g. 0 7 j) Triangular matrix: Lower Triangular matrix: A square matrix is called a lower triangular matrix if aiJ upper diagonal element should be zero . 100100 e.g. 2 2 0 0 0 0 053050 124139 e.g. 0 5 8 0 0 0 003005 200 320. upper triangular matrix : A square matrix is called an upper triangular matrix if aiJ = 0 when i < j means = 0 when i > j . k) Equality of matrics two mateices: A & B are side to be equal if they are of the same order & = bij for all i and j. each element of A is equal to the corresponding element of B , that is aiJ 1 For example: 3 5 8 are equal matrices. 58 & 3 NOTE: The number of all possible matrices of order m>< n with p number of entry is pm n example: (i)The number of all possible matrices of order 3>< 3 with entry O or 1 is 29=512. 9 (ii)The number of all possible matrices of order 3>< 3 with entry 1 or 2 or 3 is 3 03. Properties of matrix addition : Commutative property : A+B = B+A Associative property: A+ (B+C)= (A+B)+C Cancellation laws : i) Left cancellation- A+B = A+C B = C ii) Right cancellation- B+A = C+A B = C Page 2 of 7
  3. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] 04. Properties of matrix multiplication: Note that the product AB is defined only when the number of columns in matrix A is equal to the number of rows in matrix B. * If A and B are m >< n and n >< p matrices respectively then matrix AB will be an m >< p matrix i.e., order of matrix AB will be m p. * In the product AB , A is called the pre-factor and B is called the post-factor . * If the product AB is possible then it is not necessary that the product BA is also possible . * If A is a m n matrix and both AB and BA are defined then B will be a n m matrix . * If A is a n n matrix and In be the unit matrix of order n then , A In = In A-A. Matrix multiplication is associative i.e., A(BC)=(AB)C . Matrix multiplication is distributive over the addition i.e., A(B+C)=AB+AC Idempotent matrix : A square matrix A is said to be an idempotent matrix if A 100 For example 0 1 0 001 be an m n matrix, then the matrix obtained by interchanging the 05 Transpose of a Matrix : If A rows and columns of matrix A is said to be a transpose of matrix A . The transpose of A is denoted by A' or AT or AC i.e., if [aij] then AT n xm m xn [31 22 For example 0 1 6 Properties of transpose of matrices: (A+B)T = AT+ BT BT AT * (kA)T = KAT where, K is any constant 06 symmetric matrix: A square matrix A = That is, if A = [aij] then, AT = an] ah g 2 + i For example: h b f 1 1 2 3 3+2i [aij] is said to be a symmetric matrix if AT = A 3 3+2i . 4 07 Skew — symmetric matrix: A square matrix A i.e., if A= [aiJ] then AT = [ajL] 0 1 -3 02 For example 5 3 0 Fact you should know: is said to be a skew — symmetric matrix If A { Replacing j by i } (i) Note that [an ii = — [an] 2 [aii] = 0 That is, all the diagonal elements in a skew — symmetric matrix are zero. Page 3 of 7
  4. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] (ii)The matrices AAT and AT A are symmetric matrices. (iii)For any square matrix A, A+AT is a symmetric matrix and A - AT is a skew symmetric matrix always. (iv)Also any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix i.e., A= — (P) + — (Q) where P = A+AT is a symmetric matrix and Q = A- AT is a skew — symmetric matrix. 08 orthogonal matrix: A matrix A is said to be orthogonal if A.AT = I where AT is the transpose of A. 09 Invertible Matrix: If A is a squre 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 -l .A matrix having an inverse is said to be invertible . It is to note that if B is inverse of A ,then A is also the inverse of B . In other words , if it is known that AB 1 Then A-I = B —1 10. Determinants ,Minors & cofactors: a) Determinant: A unique number ( real or complex)can be associated to every square matrix A= [ao = element of order m. This number is called the determinant of the square matrix A , where aiJ of A. = det (A) and its For instance, if A = then, determinant of matrix A is written as IAI= cud value is given by ad - bc . b) Minors: Minors of an element aij of a determinant (or a determinant corresponding to matrix A ) is the determinant obtained by deleting its row and It h column in which aij lies . Minor of aij is denoted by Mij . Hence we can get 9 minors corresponding to the 9 elements of a third order (i.e.,3>
  5. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] 13. Elementary Operations or fransformations of a mafrix: The following three operations applied on the row (or column) of a matrix are called elementary row (or column) transformations- a) Interchange of any two rows (or columns): When ith row (or column) of a matrix is interchanged with the j th row (or column) , it is denoted as Ri e Rj (or Ci e Cj). b) Muläplying all elements of a row (or column) of a mafrix by a non - zero scalar: When the row (or column) of a matrix is multiplied by a scalar k, it is denoted as Ri + kRi (or Ci + kCi). c) Adding to the elements of a row (or column), the corresponding elements of any other row (or column) multiplied by any scalar k : When k times the elements of Jth row (or column) is added to the corresponding elements of the ith row (or column) , it is denoted as Ri + Ri + kR• (or Ci Ci + kC•). NOTE: In case, after applying one or more elementary row (or column) operations on A=IA (or A=AI), if we obtain all zeros in one or more rows of the matrix A on LHS , then A -1 does not exist. 14. Inverse or reciprocal of a square mafrix: If A is a square matrix of order n, then a matrix B (if such a matrix exists ) is called the inverse of A if AB=BA=In . Also note that the inverse of a matrix A is denoted by A-land we write, A —1 1. Inverse of a square matrix A exists if and only if A is non- singular matrix i.e., IAI* 0 2. If B is inverse of A then A is also the inverse of B. 15. Algorithm to find Inverse of a matrix by Elemenury Operaäons or Transformaäons: By Row Transformaäons: STEP: 1 Write the given square matrix as A=ln A STEP: 2 Perform a sequence of elementary row operations successively on A on the LHS and pre-factor In on the RHS till we obtain the result In = BA (or In = AB ). —1 STEP: 3 write A BY Column Transformaäons : STEP:I Write the given square matrix as A=A In STEP:2 Perform a sequence of elementary column operations successively on A on the LHS and post- factor In on the RHS till we obtain the result In = AB . —1 STEP:3 write A 16. Algorithm to find A-I by Determinant method: STEP: 1 Find IA! STEP:2 If IAI=O then, write "A is a singular matrix and hence not inveråble" . Else write "Ais a non - singular mafrix and hence invertible". STEP:3 Calculate the cofactors of elements of matrix A. STEP:4 Write the matrix of cofactors of elements of A and then obtain its transpose to get adj A. 1 STEP:5 Find the inverse of A by using the relation A -l = —adjA . IAI 17. Properties associated with the Inverse of mafrix & the Determinants: g)A(adjA)=(adjA)A=lAlI l) adj(AT) — (adjA)T —1 =c-1B- A h)adj(AB)=(adj B) (adj A) —1 — (adjA-l) j) (adjA) k) ladjAl=lAln I if IAI* O where n isorder of A l) IABI=IAIIBI 1 m) IAadjAl=lA In, where n is order of A -1 = —provided matrix Ais invertible o) IAI=IATI n) IA I Page 5 of 7
  6. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] IkAl = k n IAI where n is order of square matrix A and K is any scalar. If A is a non -singular matrix of order n, then ladjAl=lAlT1 1[point (k) given above]. If A is a non-singular matrix of order n, then adj(adjA) =IAITI 2 A. If A is a non-singular matrix of order n, then ladj(adjA) IA 18. Properties of Determinants: a) If any two rows or columns of a determinant are proportional or identical, then its value is zero. al bl e.g., A= a2 b2 c2 al bl [AS RI and R3 are the same] If all element of any row or column is zero , then value of determinant is zero. b) The value of a determinant remains unchanged if its rows and columns are interchanged. al bl al a2 a3 e.g., A= a2 b2 c2 Here rows and columns have been interchanged, but there is no effect on the value of determinant. c) If each element of a row or a column of a determinant is multiplied by a constant k, then the value of new determinant is k times the value of the original determinanW al bl kal kbl kC1 al bl e.g., A= a2 b2 c2 , Al = a2 d) If any two rows or columns are interchanged, then the determinant remains its absolute value , but its sign is changed. al bl e.g., A= a2 b2 c2 , Al = a2 b2 c2 al bl [Here Rl+R3]. e) If every element of some column or row is the sum of two terms, then the determinant is equal to the sum of two determinants ; one containing only the first term in place of each sum, the other only the second term . The remaining elements of both determinants are the same as given in the original determinant al + a bl e.g., A= a2 b2 c2 — (12 b2 c2 + b2 c2 D If,to each element of any row or co umn of a determinant ,the equimultiple of corresponding elements of other row ( or column ) are added, then value of determinant remains the same , i.e., the value of determinant remain same if we apply the operation Ri + Ri + kR• or Ci + Ci + kC• example: RI -+ RI + 2R2, 2 + C2 — 3C1 etc. are valid operations butR3 + 3R3 + R2 + 4C2 + Cl are not valid operations . 19. Area of triangle : Area of a triangle whose vertices are (Xl, Yl) , (x2, Y2) and (x3, h) is given by, A=- x2 1 sq, units As the area is a positive quantity, we take absolute value of the determinant given above. But if area is given &we have to find value of unknown constant(such as K),then take both positive & negative value. If the points (Xl, h) , (x2, Y2) and (x3, h) are collinear then A= O The equaäon of a line passing through the point (Xl, "1) and , (x2, Y2)can be obtained by the expression given here: X y 1 1 1 Page 6 of 7
  7. Matrix & Determinants Formulae Compiled By: Er Pawan Kumar [Maths Expert] 20. Soluåons of system of linear equaäons: a) Consistent and Inconsistent system: A System of equations is consistent if it has one or more solutions otherwise it is said to be an inconsistent system. In other words an inconsistent system of equations has no solution. b) Homogeneous and Non-homogeneous system: A system of equations AX =B is said to be a homogeneous system if B=O. Otherwise it is called a non-homogeneous system of equations. 21. Solving of system of equaåons by Mafrix method [Inverse Method] : Consider the following system of equations, al x + biy + CIZ = dl, a2X + b2Y + c2z = d2, a3X + my + c3z = d3, al bl x STEP:I Assume a2 b2 c2 , B = d2 and X = y z STEP:2 Find IAI . Now there may be following situations: a) IAI* O A -l exists . It implies that the given system of equations is consistent and therefore, the system has unique soluåon. In that case, write AX = B -11? (adjA) ] [where A Then by using the definiäon of equality of mafrice , we can get the values of x, y and z . b) IAI = O A -l does not exist. It implies that the given system of equations may be consistent or inconsistent . In order to check proceed as follow: Find (adjA)B . Now we may have either (adjA)B * 0 or (adjA) B= 0 If (adjA)B then the given system may be consistent or inconsistent . To check put z=k in the given equations and proceed ih the same manner in the new two variables system of equations assuming di — Cik , IS i < 3 as constant And if (adjA)B * O, then the given system is inconsistent with no soluåons. Page 7 of 7

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