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 a OF ARIZONA Math 215 Introduction to Linear Algebra Section 33 The Inverse of a Matrix Paul Dostert August 28 2008 112 u Definition of an Inverse Consider the scalar equation ax b where a and b are given How do we solve this equation 1 1 b axb gt ax b gtx a a a In other words we multiply both sides by the inverse of a 1a and we have an expression for ac We say 1a is the inverse of a since a 1a 1 We have a similar idea for matrices For Ax b we wish to multiply both sides by the inverse of A If A is the inverse of A then we have Axb gtAAxAb gtIxAb gtxflb where AA I If A is an n gtlt n matrix an inverse of A is an n gtlt n matrix A such that AAIampAAI If such an A exists we say A is invertible A Definition of the Inverse 5 2 3 1 Ex Show that A 5 isaninverseofA 2 Ex Show that A i i is not invertible What else do we know about this matrix Thm If A is an invertible matrix then its inverse is unique Ex Prove this Now that we have shown an inverse is unique we define the inverse of a matrix A as the matrix A l We call this A inverse Note that there is no such thing as 1A just like with vectors If you are solving a matrix equation if you want to divide by A you should multiply by A 1 instead Thm If A is an invertible n gtlt n matrix then the system of linear equations give by Ax b has the unique solution x A lb for any b e Rquot A Inverse of 2 x 2 Matrices For 2 gtlt 2 matrices we have a nice standard form for the inverse Thm If A Z b then A is invertible if ad be 3A 0 and d 1 d b Ail ad bc c a If ad be 0 then A is not invertible The quantity ad be is called the determinant of A and is denoted det A You can then think of the inverse of a 2 gtlt 2 matrix as being m times the matrix formed by switching the diagonal entries and changing the sign of the off diagonals Note This works for only 2 gtlt 2 matrices In general there is no definite form for the inverse of a matrix Ex Find the inverses of A 1 1 34 1 0B5 7and 12 3 C 4 1 ifthey exist A Properties of Invertible Matrices Thm Assume A is an invertible matrix a A 1 is invertible and Aslf1 A b If e is a nonzero scalar then CA is invertible and i 1 A 1 A 1 w G c If B is an invertible matrix the same size as A then AB is invertible and Ajay1 B lA l d AT is invertible and ATYl A1T e A is invertible for all nonnegative integers n and Ann r5 Ex Prove b and e A Properties of Invertible Matrices We can generalize property c of the previous theorem to many multiplications We have that the inverse of a product of invertible matrices is the product of the inverses in reverse order Thus A1A2 An1 A1A 1Af1 Using e we can make the following definition A 144 Anrl Note that we have a formula for the inverse of a product of matrices but not the inverse of a sum In general A B1 7 A 1 Bil Ex Give an example of matrices A and B where A B1 3A A 1 Bil Give another example where A B 1 A 1 B l Ex Solve for the matrix X assuming each operation is well defined B A2X1 A l 1421943 A Elementary Matrices 1 2 0 1 0 1 0 0 Ex ConsiderA 3 4 E1 1 0 0 and E2 0 2 0 5 6 0 0 1 0 0 1 Compute E1A and EQA Applying E1 and E2 to A results in an elementary row operation being performed on A For that reason we define an elementary matrix as a matrix that can be obtained by performing an elementary row operation on an identity matrix Ex Write E1 and E2 above in their row operation notation Ex How would we write r2 r2 27quot3 as an elementary matrix Thm Let E be the elementary matrix operation obtained by performing an elementary row operation on In If the same elementary row operation is performed on an n gtlt 7quot matrix A then the result is the same as EA Note All this theorem says is that we can write any elementary row operation as an elementary matrix multiplication A Elementary Matrices 0 1 0 1 0 0 Ex ConsiderEl 1 0 0 E2 0 2 0 and 0 0 1 0 0 1 1 0 0 E3 0 1 2 Find the inverse ofeach matrix 0 0 1 The result of the above exercise leads us to the following theorem Thm Each elementary matrix is invertible and its inverse is an elementary matrix of the same type for example the inverse of a shift is another shift Using the above theorem this means for any elementary matrix Ej we have an elementary matrix inverse E971 In particular for a product of elementary matrices we have Ek 192E1 1 Ef1E 1quot39E11 which is again the product of elementary matrices A Fundamental Thm of lnvertible Matrices The Fundamental theorem of lnvertible Matrices Version 1 Let a b C 0 6 Ex A be an n gtlt n matrix The following statements are equivalent A is invertible Ax b as a unique solution for every b e Rquot Ax 0 has only the trivial solution The rref of A is In A is a product of elementary matrices Prove this theorem using the following steps as a guide 5 C d 6 a You re going to show a a gt b has already been done and b gt c is trivial For 0 gt d figure out what the augmented system of the trivial solution looks like How do you get this as a solution For d gt e you can use the Ei to write A in its rref form Then use the inverse of the product of EZ on the previous slide For 6 gt a use the fact that the product of invertible matrices is invertible A Computing the Inverse Thm Let A be a square matrix If B is a square matrix such that AB I or BA I then A is invertible and B Ail The keyword in this theorem is or instead of and This says if we can find a matrix B with BA I or AB I then B is the inverse 1 2 3 in terms of elementary matrices Ex Write A as a product of elementary matrices What is A 1 Thm Let A be a square matrix If a sequence of elementary row operations reduces A to I then the same sequence transforms I into Ail This theorem is the key to using the GaussJordan method for computing the inverse The idea is that if you can reduce A to I then you can build I to Ail You already know how to perform Gauss Jordan to reduce A to I so you simply need to build I to A 1 at the same time 80 using elementary row operations you can do All gt IIA ll A Computing the Inverse using GaussJordan Ex If possible find the inverse of each of the following matrices Ex Solve x2y z 1 2x4z2y 4 3z3yx 1 for x x7 y z by explicitly finding the inverse first A Matlab Examples Let us compute the inverse of 2 1 0 0 0 1 2 1 0 0 A 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2 using Matlab Computing inverses is extremely easy We simply use the inv function Always check first to make sure the matrix is invertible for now by using the rref command to do this So in Matlab we could do A2 1 000 1 2 1 000 1 2 1 000 1 2 1000 1 2 rrefA invA As usual there is a nice trick you can use to type A We do this by saying we have a diagonal of size 5 consisting of 2 s then setting upper diagonal the 1 and lower diagonal the 1 to be vectors of size 4 containing 1 s A diag2ones51 diag ones411 diag ones41 1 Try to compute the inverse of a 20 gtlt 20 matrix of this form using Matlab
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