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MATRICES 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33Introduction of Matrices Tutorial Prepared By Narender Sharma Owner , Manager and Blogger Shakehand with LifePromoting quality culture in every sphere of human life. www.shakehandwithlife.blogspot.in
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Index Matrices 2S. No. Topic Page No. 1 Definition of Matrices 3-4 2 Notation of Matrices 5 3 Order of Matrices 6 4 Types of Matrices 7-11 5 Trace of Matrices 12 6 Equality of Two Matrices 13 7 MCQ‟s for Exercise 14-15 8 Answer Key 16 www.shakehandwithlife.blogspot.in
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Definition of Matrix Matrices 3A matrix is a rectangular array of numbers. In other words, a set of „m‟, „n‟numbers arranged in the form of rectangular array of m rows and n column iscalled m×n matrix read as „m‟ by „n‟ matrix. Horizontal lines a11 a12 a13 … … … . .A = a21 a22 a23 … … … . . m×n Vertical a31 a32 a33 … … … . . linesWe can understand matrix by the fig. shown above, which shows a mess ofvertical and horizontal lines. The crossing points of vertical and horizontallines are the position of elements of matrix. In above fig there are 9 crossingpoints which can be find out by multiplying no. of horizontal and verticallines i.e. 3 × 3 = 9. www.shakehandwithlife.blogspot.in
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Matrices 4The numbers a11, a12 ……..etc are called elements of the matrix A. „m‟ is thenumbers of rows and „n‟ is number of column. 2 3 4 e.g. A= 5 4 6 3×3 2 4 3 www.shakehandwithlife.blogspot.in
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Notation of Matrices Matrices 5• Matrices are denoted by capital letters A, B, C or X , Y , Z etc.• Its elements are denoted by small letters a, b, c ….. etc.• The elements of the matrix are enclosed by any of the brackets i.e. , , .• The position of the elements of a Matrix is indicated by the subscripts attached to the element. e.g. 𝑎13 indicates that element „a‟ lies in first row and third column i.e. first subscript denotes row and second subscript denote column. Element Indicate Row 𝑎1 3 Indicate Column www.shakehandwithlife.blogspot.in
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Order of Matrices Matrices 6• The number of rows and columns of a matrix determines the order of the matrix.• Hence , a matrix , having m rows and n columns is said to be of the order m × n ( read as „m‟ by „n‟).• In particular, a matrix having 3 rows and 4 columns is of the order 3 × 4 and it is called a 3 × 4 matrix e.g. 2 3 5A= is a matrix of order 2 × 3 since there are two rows and three 4 6 7 columns.A = 3 5 7 is a matrix of order 1 × 3 since there are one row and three columns. 5A = 6 is a matrix of order 3 × 1 since there are three rows and one 8 column. www.shakehandwithlife.blogspot.in
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Types of Matrices Matrices 7Rectangular Square Diagonal Scalar Matrix Matrix Matrix Matrix Column Row Null Unit Matrix Matrix Matrix Matrix Upper LowerTriangular Triangular Sub Matrix Matrix Matrix www.shakehandwithlife.blogspot.in
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Matrices 8Rectangular Matrix : A matrix in which the number of rows and columns are not equal is called a rectangular matrix e.g. , 2 4 5 A= 2 × 3 is rectangular matrix of order 2×3 4 6 7Square Matrix : A matrix in which the number of rows is equal to the number of columns is called a square matrix e.g. Principal Diagonal 2 3 4 𝐴= 3 2 5 7 8 5Note : The elements 2, 2 , 5 in the above matrix are called diagonal elements and the line alongwhich they lie is called the principal diagonal www.shakehandwithlife.blogspot.in
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Matrices 9Diagonal Matrix : A square matrix in which all diagonal elements are non- zero and all non-diagonal elements are zeros is called a diagonal matrix e.g. 2 0 0 𝐴= 0 2 0 is a diagonal matrix of 3 × 3 0 0 5 Scalar Matrix : A diagonal matrix in which diagonal elements are equal (but not equal to 1), is called a scalar matrix e.g. 2 0 0 𝐴= 0 2 0 is a scalar matrix of 3 × 3 0 0 2Identity (or Unit) Matrix : A square matrix whose each diagonal element is unity and all other elements are zero is called and Identity (or Unit) Matrix. An Identity matrix of order 3 is denoted by I3 or simply by I. www.shakehandwithlife.blogspot.in
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Matrices 10e.g. 1 0 0 I3 = 0 1 0 is a unit matrix of order 3 0 0 1Null (Zero) Matrix : A matrix of any order (rectangular or square) whose each of its element is zero is called a null matrix (or a Zero matrix) and is denoted by O. e.g. 0 0 0 0 0 O= and O = are null matrices of order 2 × 2 and 2 × 3 0 0 0 0 0respectively.Row Matrix : A matrix having only one row and any number of columns is called a row matrix (or a row vector) e.g. 𝐴= 1 2 3 is a row matrix of order 1 × 3 www.shakehandwithlife.blogspot.in
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Matrices 11 Column Matrix : A matrix having only one column and any number of rows is called a column matrix (or a column vector) e.g. 1 A = 3 is a column matrix or order 3 × 1 6 Upper Triangular and Lower Triangular Matrix: A square matrix is called an upper triangular matrix if all the elements below the principal diagonal are zero and it is said to be lower triangular matrix if all the elements above the principal diagonal are zero e.g. UTM 2 3 4 LTM 5 0 0 𝐴= 0 1 5 , 𝐵= 8 7 0 0 0 7 4 3 1Sub Matrix : A matrix obtained by deleting some rows or column or both of a given matrix is called its sub matrix. e.g. 2 3 4 2 3 𝐴 = 5 1 5 , Now is a sub matrix of given matrix A. The sub 3 9 3 9 7matrix obtained by deleting 2nd row and 3rd column of matrix A. www.shakehandwithlife.blogspot.in
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Trace of Matrix Matrices 12The sum of all the elements on the principal diagonal of a square matrix iscalled the trace of the matrix. It is denoted by tr. A . a11 a12 a13 If A= a21 a22 a23 then a31 a32 a33 trace of A = tr.A = 𝑎11 + 𝑎22 + 𝑎33 2 3 4 e.g. A = 5 1 5 , then 3 9 7 trace of A = tr. A =2+1+7=10 www.shakehandwithlife.blogspot.in
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Equality of Two Matrices Matrices 13Two matrices „A‟ and „B‟ are said to be equal if only if they are of the sameorder and each element of „A‟ is equal to the corresponding element of „B‟.Given a11 a12 a13…….. b11 b12 b13………A = a21 a22 a23…….. m×n and B = b21 b22 b23……… x×y a31 a32 a33…….. b31 b32 b33………The above two matrices will be equal if and only if No. of rows and column of A and B are equal i.e. m=x and n=y. Each element of A is equal to the corresponding element of B i.e. a11 =b11 , a12 = b12 , a13 = b13 ……… and so on 𝑎 𝑏 3 4Let 𝐴 = and 𝐵 = , according to the condition of equality of 𝑐 𝑑 5 6matrices both matrices have same order i.e. 2 × 2 but the 2nd necessarycondition is a=3, b=4, c=5 , d= 6 www.shakehandwithlife.blogspot.in
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MCQ’s for Excercise Matrices 141. If in a given matrix the rows are 2 and the column are 3 then maximum no. of elements in thismatrix areA) 4 B) 6 C) 8 D) 102. Element a32 is belongs to which row and column in a given matrix AA) 2nd row, 3rd columnB) 3rd row , 2nd columnC) can’t decideD) None of the above 2 4 5 5 73. What is the order of given matrix 7 5 7 8 1 9 0 3 2 4A) 3 × 5 B) 5 × 3 C) 3 × 3 D) 5 × 54. A matrix which do not have equal no. of rows and columns calledA) Square Matrix B) Triangular Matrix C) Scalar Matrix D) Rectangular Matrix5. A matrix in which the rows and columns are equal in no. is calledA) Square Matrix B) Triangular Matrix C) Scalar Matrix D) Rectangular Matrix6. In a Diagonal matrix all the elements on the principal diagonal can’t be equal toA) 0 B) 1 C) 2 D) 3 www.shakehandwithlife.blogspot.in
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Matrices 157. In a scalar matrix all the diagonal elements areA) Equal to one B) Equal but greater than one C) Not equal D) None of the above8. In a Unit matrixA) All element are zero B) All elements are one C) Diagonal elements zero rest areone D) Diagonal elements are one rest are zero9. In a given matrix all elements below principal diagonal are zero , the matrix isA) Upper triangular matrix B) Lower triangular matrix C) Diagonal Matrix D) Scalar Matrix10. In a given matrix all elements above principal diagonal are zero , the matrix isA) Upper triangular matrix B) Lower triangular matrix C) Diagonal Matrix D) Scalar Matrix a11 a12 a1311. Given A = a21 a22 a23 then tr. A will be a31 a32 a33A) 𝑎11 + 𝑎12 + 𝑎13 B) 𝑎11 + 𝑎22 + 𝑎33 C) 𝑎31 + 𝑎22 + 𝑎13 D) 𝑎21 + 𝑎22 + 𝑎2312. Which one is true about the necessary condition for equality of two matricesA) Order of two matrices are sameB) Corresponding elements of the two matrices are sameC) Both A and BD) None of the above www.shakehandwithlife.blogspot.in
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Answer Key Matrices 16S.No. 1 2 3 4 5 6Ans. B B A D A AS.No. 7 8 9 10 11 12Ans. B D A B B C To get free copy of this tutorial E mail: shakehandwithlife@gmail.com Mention your name, place , profession with Subject : “Tutorial - Introduction of Matrices” For more E- Tutorial , Notes , Books, previous solved papers, Educational and motivational articles , and related documents Visit : www.shakehandwithlife.blogspot.in Like us on Facebook Page : Shakehand with Life www.shakehandwithlife.blogspot.in
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