Matrices: Definitions, Types, and Operations PDF

## MATRICES ### 2 DEFINITION - If A, B, C be any three cities such that there are three ways to go from A to B, one way to go from A to C, and no way to go from A to A. Similarly, there are three ways to go from B to A, no way to go from B to B, two ways to go from B to C, and one way to go from C...

## MATRICES ### 2 DEFINITION - If A, B, C be any three cities such that there are three ways to go from A to B, one way to go from A to C, and no way to go from A to A. Similarly, there are three ways to go from B to A, no way to go from B to B, two ways to go from B to C, and one way to go from C to A, two ways to go from C to B, and one way to go from C to C. Then, it can be shown in the following form with the help of rows and columns. ``` ABC A 0 3 1 B 3 0 2 C 1 2 1 ``` This is a matrix. So, the definition of a matrix is: An arrangement of numbers in the form of a rectangular array having, m rows and n columns, is called an mxn matrix. The general form of a mxn matrix is given below, ``` a11 a12 ... ain a21 a22 ... a2n ... ... ... ... am1 am2 ... amn ``` Here the numbers a11, a12 ..., amn are called "elements" of the matrix. aij is the general element which is placed in ith row and jth column of the matrix. In a matrix, the horizontal lines are called "rows or row vectors" and vertical lines are called "columns or column vectors". Also mxn is called the "order or the type" of the matrix. ### 3 TYPES OF MATRICES - - **(1) Horizontal Matrix** - In this type of matrix, the number of rows is less than the number of columns. For example: ``` 3 4 7 5 2 4 ``` - **(2) Vertical Matrix** - In this type of matrix, the number of columns is less than the number of rows. For example: ``` 2 3 4 5 4 7 0 1 9 2 8 6 ``` - **(3) Row Matrix or Row Vector** - This type of matrix has only one row, and any number of columns (say n). For example: ``` [a1 a2 a3], [3, 2], [a11 a12... ain] ``` - **(4) Column Matrix or Column Vector** - This type of matrix has only one column, and any number of rows (says n). For example: ``` a11 a21 ... am1 ``` ``` a1 b1 c1 ``` - **(5) Square Matrix** - In a square matrix, the number of rows is equal to the number of columns. For example: ``` b1 b2 b3 c1 c2 c3 ``` ``` 2 6 4 1 3 3 3 4 4 9 7 2 1 5 2 3 ``` - **(6) Zero or Null Matrix** - A matrix in which each element is equal to zero, is called a "zero or a null matrix". It is denoted by 0. For example: ``` 0 0 0 0 0 0 0 0 0 0 0 0 ``` ``` [0 0 0] ``` - **(7) Diagonal Matrix** - A square matrix in which each element except the leading diagonal elements is equal to zero, is called a "diagonal matrix." For example: ``` [3 0 0 ] [0 1 0 ] [0 0 1 ] ``` ``` 2 0 0 0 1 0 0 0 1 ``` - **(8) Scalar Matrix** - A square matrix in which all the leading diagonal elements are equal, and all other elements are zero, is called a "scalar matrix." For example: ``` a 0 0 0 a 0 0 0 a ``` ``` 2 0 0 0 2 0 0 0 2 ``` - **(9) Unit or Identity Matrix** - A Square matrix, in which each leading diagonal element is unity, and all other elements are zero, is called a unit or identity matrix. For example: ``` 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 ``` ``` 0 0 1 0 0 0 0 1 ``` ### 4 EQUAL MATRICES – The two matrices A = [aij] and B = [bij] will be equal, if - (i) Numbers of rows and columns are the same for A and B i.e. A and B are of same order (or type), and - (ii) The corresponding elements of A and B are equal i.e. aij = bij (for every i and j). ### 5 ADDITION OF MATRICES - If A = [aijlm×n and B = [bijlm×n be any two matrices of the same order mxn, then the matrix A + B is mxn matrix obtained by adding corresponding elements of A and B. A + B = [aij]m×n + [bij]m×n = [aij + bijlm×n ### 6 SUBTRACTION OF MATRICES - If A and B be any two m × n matrices, then the matrix A – B is obtained by subtracting their corresponding elements. A − B = A + (− B) = [aij]m×n − [bij]m×n ### 7 MULTIPLICATION OF MATRIX BY A SCALAR - To find scalar multiple of a matrix A by k, each element of matrix A is multiplied by the scalar k. If A = [aij], then kA = [kaj] ### 8 MULTIPLICATION OF MATRICES – The product AB of two matrices A and B is possible only if the number of columns in matrix A is equal to the number of rows in matrix B. If A be any matrix of type mxp and B is any matrix of type pxn then the product AB = C shall be a matrix of type m × n. ### 8.1 PROPERTIES OF MATRIX MULTIPLICATION - - (1) Matrix multiplication is associative: A (BC) = (AB) C - (2) Matrix multiplication is distributive over matrix addition: A (B + C) = AB + AC - (3) A.1=1.A = A where A is a square matrix of order n x n and I is a unit matrix of order n x n - (4) A² = A.A; A³ = A .A .A and in general Aº = A . A.A to n factors. ### 9 TRANSPOSE OF A MATRIX - The matrix, obtained by interchanging the rows into columns and columns into rows of a given matrix A, is called "Transpose of A" and is denoted by A' or A^T. ### 10 PROPERTIES OF TRANSPOSE OF A MATRIX - (1) (A')' = A - (2) (A + B)' = A' + B' - (3) (AB)' = B' A' - (4) (CA)' = CA', where C is any scalar ### 11 SYMMETRIC MATRIX - A square matrix A is called a symmetric matrix if A = A'. ### 12 SKEW-SYMMETRIC MATRIX - If A is a square matrix and A' = - A, then the square matrix is called a skew-symmetric matrix. ### 13 DETERMINANT OF A SQUARE MATRIX - If A = [aij]n × n is a square matrix, then the determinant of A is det A or |A| = ### 14 SINGULAR MATRIX A square matrix, the value of whose determinant is zero, is called a "singular matrix." ### 15 NON-SINGULAR MATRIX A square matrix value of whose determinant is not zero, is called "non-singular matrix." For example, A = is a non-singular matrix, since |A| = = 36 –-35 = 1 ≠ 0. ### 16 ADJOINT MATRIX If A = [aij]n × n be a square matrix, then the matrix, obtained by replacing each element aij of A by its cofactor Aij, is denoted by [Aij], and the transpose of this matrix [Aij] is called "adjoint of matrix A" and denoted by 'adj; A'. ### 17 PROPERTIES OF ADJOINT OF A MATRIX - (1) A. (adj A) = (adjA) .A = |A | .I where I is a unit matrix of order n x n. - (2) adj (AB) = (adj B). (adj A) ### 18 INVERSE OF A MATRIX OR INVERSE MATRIX If the product of a matrix A with a matrix B is equal to a unit matrix 1, then B is called "inverse matrix" of A, and it is denoted by A^-1. i.e. A^-1A = AA^-1 = I. ### 19 PROPERTIES OF AN INVERSE MATRIX - (1) (A^-1)^-1 = A - (2) (AB)^-1 = B^-1A^-1 where A and B are non-singular square matrices of the same order. ### 20 METHOD OF FINDING THE INVERSE MATRIX If A be any non-singular matrix i.e. |A | ≠ 0, then A^-1 = (Adj A)/|A|. ### 21 ELEMENTARY TRANSFORMATIONS The following operations on a matrix are called elementary transformations, E-transformations, or operations. - (1) Interchange of two rows or two columns. Interchange of the ith and ith rows is denoted by Rij. Interchange of ith and ith columns is denoted by Cij. - (2) Multiplication of any row or column by a non-zero number k denoted by k R₁ or k C₁. - (3) Addition k times the ith row, to jth row or k times the ith column to jth column, denoted by Rj + kR; or Cj + kC;. ### 22 ELEMENTARY MATRICES The matrix obtained from a unit matrix I by a single E-operation is called an elementary matrix. ### 23 DETERMINATION OF RANK OF A MATRIX - Method I-By Converting the given matrix into Triangular form - - Method II-By Converting the matrix into Normal Form - - Method III - ### 24 SOLUTION OF SYSTEM OF LINEAR EQUATION BY MATRIX METHOD Let the equation be ax + azy + az = d₁ bx + by + bzz = d₂ c₁x + czy + czy = d3 ### 25 SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS BY MATRIX METHOD