Matrices & Matrix

Matrices & Matrix Summary A matrix is defined as a rectangular array of elements. If the arrangement has m rows and n columns, then the matrix is of order mxn (read as m by n). A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter. Two matrices are said to be comparable if they have the same order. Addition and subtraction of two matrices is possible only if they have the same order. If two matrices A and B are of same order, then A - B = A + (- B). Commutative law, associative law holds good for addition of matrices. The additive identity of a matrix A of order mxn is the zero matrix of order mxn. The additive inverse of a matrix A is -A. The multiplication of two matrices A and B is possible if the number of columns of A is equal to the number of rows B. Suppose A is a matrix of order mxn and B is a matrix of order nxp, the matrix AB is of order mxp. If A, B and C are the matrices which can be multiplied then (a) Matrix multiplication is not commutative, i.e., AB BA (always) (b) Associative law holds good for matrix multiplication, i.e., (AB)C = A(BC) (c) Matrix multiplication is distributive with respect to addition A(B + C) = AB + AC or (A + B)C = AC + BC If A is a matrix of order mxn and is a scalar (real or complex) then the matrix kA is obtained by multiplying each element of A by k. To every square matrix, a value can be associated which is known as the determinant of the matrix. Note that the determinant of kA where k is a scalar and A is a square matrix, is given by kn times determinant of A. i.e., is |kA| = kn |A| The value of the determinant remain unchanged if its rows and columns are interchanged If two rows or columns of a determinant are interchanged, then the sign of the determinant is changed. If any two rows or columns of a determinant are equal, then its value is zero. If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k. If two rows or columns of determinant are proportional, the value of the determinant is zero. A square A = [aij] is said to be symmetric if AT = A, i.e., if aij = aji A square matrix A is said skew symmetric if AT = - A, i.e., aij = - aji Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix as follows For a 2 x 2 matrix, the adjoint is got by interchanging elements in the leading diagonal and changing signs in the other diagonal. If A =[aij]mxn is a matrix of order mxn. The minor of aij of |A|, denoted by Mij, is given by the determinant which is obtained by deleting ith row jth column of |A|. The co-factor of the determinant of the A = [aij]mxn, denoted by Aij is given by Aij = (-1)i+j Mij The transpose of a matrix A, denoted by AT, is obtained by interchanging the rows and columns of A. The adjoint of a square matrix A = [aij] is defined as the transpose of the matrix [Aij] where Aij is the co-factor of the element aij. Adjoint of A is denoted by Adj A. Note that the concept of adj is only for square matrix. A square matrix A is said to be non-singular if |A| 0. Let A be a square matrix of order n. If there exists a square matrix B of order n, such that AB = BA = In, where In is the identify matrix of order n, then B is called the inverse of A. The inverse of a matrix A exists if and only if |A| 0. In other words, every non-singular matrix is invertible. The area of a triangle whose vertices are (x1, y1), (x2, y2) and The following are the steps to solve a system of linear equations using Cramer's rule. Step 1: Find the value of the determinant Step 2: If D 0, then the system has unique solution, given by Where D1, D2 and D3 are the determinants obtained from D by replacing respectively the first column, 2nd column and third column containing the constant terms d1, d2, d3. Step 3: If D = 0, the system may have infinite number of solutions or no solution. A system of linear equations is said to be consistent if it has at least are solution, otherwise it is inconsistent. Let A be asquare matrix of order n. Following are the steps to find the inverse of a matrix. Step 1: Find the value of the determinants A. That is, find |A| Step 2: If |A| = 0, inverse of the matrix A does not exists. Step 3: If |A| 0, find the co-factors Aij of all the elements of A. Step 4: Find adj A, the transpose of the matrix of co-factors Aij. Step 5: Following are the steps to solve a system of linear equations with three unknown, using inverse of a matrix (Matrix method) Let the given system of equations be Step 1: The system of linear equations may be expressed as AX = B. Step 2: Find |A|. If |A| 0, the system has unique solution which is given by X = A-1B. Step 3: If |A| = 0, put x = k (y = k or z = k) in any two of the given equations and find y and z in terms of k. Substitute these values of x, y and z in terms of k in the third equation. If the third equation is satisfied by these values of x, y and z, then the system has infinitely many solutions. If the third equation is not satisfied, the system has no solution