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Transcript
Matrices and Determinants Page 1 of 10 On this page will be: Introduction and Examples Matrix Addition and Subtraction Matrix Multiplication The Transpose of a Matrix The Determinant of a Matrix The Inverse of Matrix Systems of Linear Equations The Inverse Matrix Method Cramer's Rule Introduction and Examples DEFINITION: A matrix is defined as an ordered rectangular array of numbers. They can be used to represent systems of linear equations, as will be explained below Here are a couple of examples of different types of matrices: Symmetric Diagonal Upper Triangular Lower Triangular Zero Identity And a fully expanded mxn matrix A, would look like this: or in a more compact form: Matrix Addition and Subtraction DEFINITION: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of rows and columns. Take: http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 2 of 10 Addition If A and B above are matrices of the same type then the sum is found by adding the corresponding elements aij+bij Here is an example of adding A and B together Subtraction If A and B are matrices of the same type then the subtraction is found by subtracting the corresponding elements aij-bij Here is an example of subtracting matrices Now, try adding and subtracting your own matrices Matrix Multiplication DEFINITION: When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed. Here is an example of matrix multiplication for two 2x2 matrices Here is an example of matrices multiplication for a 3x3 matrix Now lets look at the nxn matrix case, Where A has dimensions mxn, B has dimensions nxp. Then the product of A and B is the matrix C, which has dimensions mxp. The ijth element of http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 3 of 10 matrix C is found by multiplying the entries of the ith row of A with the corresponding entries in the jth column of B and summing the n terms. The elements of C are: Note: That AxB is not the same as BxA Now, try multiplying your own matrices Transpose of Matrices DEFINITION: The transpose of a matrix is found by exchanging rows for columns i.e. Matrix A = (aij) and the transpose of A is: AT=(aji) where j is the column number and i is the row number of matrix A. For example, The transpose of a matrix would be: In the case of a square matrix (m=n), the transpose can be used to check if a matrix is symmetric. For a symmetric matrix A = AT Now try an example The Determinant of a Matrix DEFINITION: Determinants play an important role in finding the inverse of a matrix and also in solving systems of linear equations. In the following we assume we have a http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 4 of 10 square matrix (m=n). The determinant of a matrix A will be denoted by det(A) or |A|. Firstly the determinant of a 2x2 and 3x3 matrix will be introduced then the nxn case will be shown. Determinant of a 2x2 matrix Assuming A is an arbitrary 2x2 matrix A, where the elements are given by: then the determinant of a this matrix is as follows: Now try an example of finding the determinant of a 2x2 matrix yourself Determinant of a 3x3 matrix The determinant of a 3x3 matrix is a little more tricky and is found as follows ( for this case assume A is an arbitrary 3x3 matrix A, where the elements are given below) then the determinant of a this matrix is as follows: Now try an example of finding the determinant of a 3x3 matrix yourself Determinant of a nxn matrix For the general case, where A is an nxn matrix the determinant is given by: Where the coefficients where are given by the relation is the determinant of the (n-1) x (n-1) matrix that is obtained by deleting row i and column j. This coefficient is also called the cofactor of aij. http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 5 of 10 The Inverse of a Matrix DEFINITION: Assuming we have a square matrix A, which is non-singular ( i.e. det(A) does not equal zero ), then there exists an nxn matrix A-1 which is called the inverse of A, such that this property holds: AA-1= A-1A = I where I is the identity matrix. The inverse of a 2x2 matrix Take for example a arbitury 2x2 Matrix A whose determinant (ad-bc) is not equal to zero where a,b,c,d are numbers, The inverse is: Now try finding the inverse of your own 2x2 matrices: The inverse of a nxn matrix The inverse of a general nxn matrix A can be found by using the following equation: Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be calculated by the following method Given the nxn matrix A, define to be the matrix whose coefficients are found by taking the determinant of the (n-1) x (n-1) matrix obtained by deleting the ith row and jth column of A. The terms of B (i.e. B = bij) are known as the cofactors of A. And define the matrix C, where . The transpose of C (i.e CT) is called the adjoint of matrix A. Lastly to find the inverse of A divide the matrix CT by the determinant of A to give its inverse. http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 6 of 10 Now test this method with finding the inverse of your own 3x3 matrices Solving Systems of Equations using Matrices DEFINITION: A system of linear equations is a set of equations with n equations and n unknowns, is of the form of The unknowns are denoted by x1,x2,...xn and the coefficients (a's and b's above) are assumed to be given. In matrix form the system of equations above can be written as: A simplified way of writing above is like this ; Ax = b Now, try putting your own equations into matrix form by clicking here: After looking at this we will now look at two methods used to solve matrices these are Inverse Matrix Method Cramer's Rule Inverse Matrix Method DEFINITION: The inverse matrix method uses the inverse of a matrix to help solve a system of equations, such like the above Ax = b. By pre-multiplying both sides of this equation by A-1 gives: http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 7 of 10 or alternatively this gives So by calculating the inverse of the matrix and multiplying this by the vector b we can find the solution to the system of equations directly. And from earlier we found that the inverse is given by From the above it is clear that the existence of a solution depends on the value of the determinant of A. There are three cases: 1. If the det(A) does not equal zero then solutions exist using 2. If the det(A) is zero and b=0 then the solution will be not be unique or does not exist. 3. If the det(A) is zero and b=0 then the solution can be x = 0 but as in 2. is not unique or does not exist. Looking at two equations we might have that Written in matrix form would look like and by rearranging we would get that the solution would look like Now try solving your own two equations with two unknowns Similarly for three simultaneous equations we would have: http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 8 of 10 Written in matrix form would look like and by rearranging we would get that the solution would look like Now try solving your own three equations with three unknowns Cramer's Rule DEFINITION: Cramer's rules uses a method of determinants to solve systems of equations. Starting with equation below, The first term x1 above can be found by replacing the first column of A by . Doing this we obtain: Similarly for the general case for solving xr we replace the rth column of A by and expand the determinant. http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 9 of 10 This method of using determinants can be applied to solve systems of linear equations. We will illustrate this for solving two simultaneous equations in x and y and three equations with 3 unknowns x, y and z. Two simultaneous equations in x and y To solve use the following: and or simplified: and Now try solving two of your own equations Three simultaneous equations in x, y and z ax+by+cz = p dx+ey+fz = q gx+hy+iz = r To solve use the following: , and http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009 Matrices and Determinants Page 10 of 10 Now try solving your own three equations back forward http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html 5/14/2009