Class Note for MATH 215 with Professor Dostert at UA
Class Note for MATH 215 with Professor Dostert at UA
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Date Created: 02/06/15
THE UNIVERSITY Q OF ARIZONA Math 215 Introduction to Linear Algebra Section 42 Determinants Paul Dostert September 23 2008 110 A Determinant of a 2 x 2 Matrix Recall we have previously defined the determinant of a 2 gtlt 2 matrix A all Q12 as det A a11a22 a12a21 How was this use in the 21 022 formula for the inverse of a 2 gtlt 2 matrix The determinant of any matrix A is usually denoted by either A or det A We may also write 011 012 A l a a 21 22 We essentially replace the bracket notation around the matrix with absolute value type notation indicated we want the determinant of the matrix We define the determinant of a 1 gtlt 1 matrix A as detA a a where a does NOT indicate absolute value A Determinant of a 3 x 3 Matrix The determinant of a 3 gtlt 3 matrix can be thought of as a linear combination of determinants of 2 gtlt 2 matrices The process is exactly the same as the cross product 011 012 013 Let A Q21 G22 G23 031 032 033 Then the determinant of A is the scalar G22 a G32 a Q21 a det A G11 Q31 a G21 G22 I 23 23 G12 13 33 33 G31 032 We define the Lj minor of A AZj as the submatrix of A obtained by deleting row 139 and column j We may then write det A G11 det A11 Q12 det A12 G13 det A13 Ex Find the determinant of A Determinant of an n x n Matrix Let A aij be an n gtlt n matrix n 2 2 Then the determinant of A is det A all det A11 G12 det A12 13 det A13 aln det A1 The minor with its sign is the Lj cofactor ofA Then we may write det A Z ale39lj j1 Interestingly the determinant can be computed similarly using any other row Laplace Extension Thm An n gtlt n matrix A aij n 2 2 has I L I L detA E aijC39Z39j E aZjC39Z39j j1 z391 These are the cofactor expansion along the ith row and jth column respectively A Computing determinants Ex The previous theorem tells us that we may choose any row or column to compute the determinant along Use this idea to find the determinant of each of the following 4 0 1 i 1 1 1 3 1 0 0 0 1 7 7 0 0 1 1 3 1 1 0 1 0 0 1 2 1 1 1 0 1 1 1 Ex Show a b C b a C a 0 1 2 A Properties of determinants Thm Let A AZj be a square matrix a If A has a zero row column then det A 0 b If B is obtained by interchanging two rows columns of A then det B det A c If A has two identical rows columns then det A 0 d If B is obtained by multiplying a row column of A by k then det B ldet A e If A B and C39 are identical except that the ith row column of C39 is the sum of the ith rows columns of A and B then det G det A det B f If B is obtained by adding a multiple of one row column of A to another row column then det B det A 2 2 4 Ex Compute det A for 6 2 1 3 1 2 Ex Compute det A for 6 2 0 by reducing to rref Remember to keep 0 1 4 track of scalars A Determinants of Elementary Matrices Thm Let E be an n gtlt n elementary matrix a If E results from interchanging two rows of In then detE 1 b If E results from multiplying one row or In by k then det E k c If E results from adding a multiple of one row of In to another row then det E I Lemma Let B be an n gtlt n matrix and E be an n gtlt n elementary matrix Then det det E det B The previous theorem and lemma can be used to prove the following Thm A square matrix A is invertible iff det A 7 0 Pf The idea is the you reduce A to rref R using elementary matrices E1 7 ET Write ErE2E1AR and take the determinant of both sides A Determinants and Matrix Operations Thm If A is an n gtlt n matrix then det kA kquot det A Note In general there is no formula for det A B Thm If A and B are n gtlt n matrices then det det A det B i 1 2 i 4 1 ExForA1 2andB1 1 AB and finding its determinant and then by using the theorem find det AB by first calculating Thm If A is invertible then 1 detA1 detA Ex Find det A l for A from the previous exercise by explicitly finding A 1 and by using the previous theorem A Aside Cramer s Rule and the Adjoint The book presents Cramer s Rule which allows you to solve small matrix problems easily The idea is that we can find the solution to a linear system simply from computing determinants The text also presents the adjoint of a matrix These can both be used to help prove the Laplace Expansion Theorem Their practical application is limited thus we will discuss these in class only if time allows If you are a Math major make sure you read through each of these concepts The last theorem for this chapter is used as part of the proof of the Laplace Expansion Theorem Thm Let A be an n gtlt n matrix and let B be obtained by interchanging any two rows columns of A Then detB det A A Matlab Determinants As with most subjects in linear algebra Matlab has a very intuitive way to compute the determinant We simply define a matrix and apply the function det To find the determinant of Al 00000 l l l l Axon we do gtgt A 312310014 detA Let us say we wish to verify det A10 det A1O We do gtgt A 312310014 detAquot10 detAquot10
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