MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms M Matrices and Linear Algebra 1 Matrix algebra In section D we calculated the determinants of square arrays of numbers Such arrays are important in mathematics and its applications they are called matrices In general they need not be square only rectangular A rectangular array of numbers having m rows and n columns is called an m x n matrix The number in the i th row and j th column where 1 5 i 5 m 1 5 j 5 n is called the ij entry and denoted aij the matrix itself is denoted by A or sometimes by aij Two matrices of the same size are equal if corresponding entries are equal Two special kinds of matrices are the row vectors the 1 x n matrices a l az a and the column vectors the m x 1 matrices consisting of a column of m numbers From now on row vectors or column vectors will be indicated by boldface small letters when writing them by hand put an arrow over the symbol Matrix operations There are four basic operations which produce new matrices from old 1 Scalar multiplication Multiply each entry by c cA caij 2 Matrix addition Add the corresponding entries A B aij bij the two matrices must have the same number of rows and the same number of columns 3 Transposition The transpose of the m x n matrix A is the n x m matrix obtained by making the rows of A the columns of the new matrix Common notations for the transpose are AT and A using the first we can write its definition as AT aji If the matrix A is square you can think of AT as the matrix obtained by flipping A over around its main diagonal 2 3 Example 1 1 Let A 1 5 F i n d A B AT 2A 3B 18 02 NOTES 2 4 Matrix multiplication This is the most important operation Schematically we have mxn nxp mxP The essential points are 1 For the multiplication to be defined A must have as many columns as B has rows 2 The ij th entry of the product matrix C is the dot product of the i th row of A with the j th column of B The two most important types of multiplication for multivariable calculus and differential equations are 1 AB where A and B are two square matrices of the same size these can always be multiplied 2 Ab where A is a square n x n matrix and b is a column n vector Laws and properties of matrix multiplication M 1 A B C AB AC M 2 AB C A BC A B C AC B C cA B c AB distributive laws associative laws In both cases the matrices must have compatible dimensions M 3 Let I3 then A I A and I A A for any 3 x 3 matrix I is called the identity matrix of order 3 There is an analogously defined square identity matrix Inof any order n obeying the same multiplication laws M 4 In general for two square n x n matrices A and B AB BA matrix multiplication is not commutative There are a few important exceptions but they are very special for example the equality A I I A where I is the identity matrix M 5 For two square n x n matrices A and B we have the determinant law lABl IAIJBI also written det AB det A det B For 2 x 2 matrices this can be verified by direct calculation but this naive method is unsuitable for larger matrices it s better to use some theory We will simply assume it in these notes we will also assume the other results above of which only the associative law M 2 offers any difficulty in the proof M MATRICES AND LINEAR ALGEBRA 3 M 6 A useful fact is this matrix multiplication can be used to pick out a row or column of a given matrix you multiply by a simple row or column vector to do this Two examples should give the idea i g 8 i 1 0 0 4 5 6 1 the second column 2 3 thefirstrow Exercises Section 1F 2 Solving square systems of linear equations inverse matrices Linear algebra is essentially about solving systems of linear equations an important application of mathematics to real world problems in engineering business and science especially the social sciences Here we will just stick to the most important case where the system is square i e there are as many variables as there are equations In low dimensions such systems look as follows we give a 2 x 2 system and a 3 x 3 system In these systems the aij and bi are given and we want to solve for the xi As a simple mathematical example consider the linear change of coordinates given by the equations If we know the y coordinates of a point then these equations tell us its x coordinates immediately But if instead we are given the x coordinates to find the y coordinates we must solve a system of equations like 7 above with the yi as the unknowns Using matrix multiplication we can abbreviate the system on the right in 7 by where A is the square matrix of coefficients a i j The 2 x 2 system and the n x n system would be written analogously all of them are abbreviated by the same equation Ax b notice You have had experience with solving small systems like 7 by elimination multiplying the equations by constants and subtracting them from each other the purpose being to 18 02 NOTES 4 eliminate all the variables but one When elimination is done systematically it is an efficient method Here however we want to talk about another method more compatible with handheld calculators and MatLab and which leads more rapidly to certain key ideas and results in linear algebra Inverse matrices Referring to the system 8 suppose we can find a square matrix M the same size as A such that MA I 9 the identity matrix We can then solve 8 by matrix multiplication using the successive steps where the step M Ax x is justified by M A x MA x Ix x by M 2 by 9 by M 3 Moreover the solution is unique since 10 gives an explicit formula for it The same procedure solves the problem of determining the inverse to the linear change of coordinates x Ay as the next example illustrates 1 2 2 Verify that M satisfies 9 2 3 and M 32 1 above and use it to solve the first system below for xi and the second for the yi in terms of the xi E x a m p l e 2 1 Let A Solution 1 2 We have 2 3 3 2 solve the first …
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