Solving Linear Systems, Continued And The Inverse Of A Matrix

Solving LinearSystemsMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionSolving Linear Systems, ContinuedandThe Inverse of a MatrixMath 240 — Calculus IIISummer 2015, Session IITuesday, July 7, 2015

Solving LinearSystemsAgendaMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusion1. Solving Linear SystemsGauss-Jordan eliminationThe rank of a matrix2. The inverse of a square matrixDefinitionComputing inversesProperties of inversesUsing inverse matricesConclusion

Solving LinearSystemsGaussian eliminationMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionGaussian elimination solves a linear system by reducing toREF via elementary row ops and then using back substitution.Example3x1 2x2 x1 2x2 2x1 x2 1 002x3 9x3 52x3 1 210131 3 229 1 215 2 1 2 1 5x1 2x2 x3 55 x2 3x3 52x3 2Steps1. P123. A13 ( 2)5. A23 ( 3) 2. A12 ( 3)4. A32 ( 1)6. M3 113Back substitution gives the solution (1, 1, 2).

Solving LinearSystemsGauss-Jordan eliminationMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionReducing the augmented matrix to RREF makes the systemeven easier to solve.Example 1 2 1 51 01 3 5 000 1 20010 0 10 1 1 2x1x2 1 1x3 2Steps1. A32 ( 3)2. A31 ( 1)3. A21 (2)Now, without any back substitution, we can see that thesolution is (1, 1, 2).The method of solving a linear system by reducing itsaugmented matrix to RREF is called Gauss-Jordanelimination.

Solving LinearSystemsThe rank of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionDefinitionThe rank of a matrix, A, is the number of nonzero rows it hasafter reduction to REF. It is denoted by rank(A).If A is the coefficient matrix of an m n linear system andrank(A# ) rank(A) n then the REF looks like 1 ··· x1 1 . . . x2 . . . . 1 xn 0 . 00LemmaSuppose Ax b is an m n linear system with augmentedmatrix A# . If rank(A# ) rank(A) n then the system has aunique solution.

Solving LinearSystemsThe rank of a matrixMath 240Solving ermine the solution set of the linear systemx1 x2 x3 x4 1,2x1 3x2 x3 4,3x1 5x2 3x3 x4 ties ofinversesUsing inversematricesConclusionReduce the augmented matrix. 1 1 11 1 A12 ( 2) 1 1 1 1 1A13 ( 3) 2 310 4 0 1 3 2 2 A23 ( 2)3 53 1 50 0 0 0 2The last row says 0 2; the system is inconsistent.LemmaSuppose Ax b is a linear system with augmented matrixA# . If rank(A# ) rank(A) then the system is inconsistent.

Solving LinearSystemsThe rank of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionExampleDetermine the solution set of the linear system5x1 6x2 x3 4,2x1 3x2 x3 1,4x1 3x2 x3 5.Reduce the augmented matrix. 5 61 41 0 1 2 31 1 0 1 14 3 1 50 0 0 21 0x1 x3 2x2 x3 1The unknown x3 can assume any value. Let x3 t. Then byback substitution we get x2 t 1 and x1 t 2. Thus, thesolution set is the line{(t 2, t 1, t) : t R} .

Solving LinearSystemsThe rank of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionDefinitionWhen an unknown variable in a linear system is free to assumeany value, we call it a free variable. Variables that are not freeare called bound variables.The value of a bound variable is uniquely determined by achoice of values for all of the free variables in the system.LemmaSuppose Ax b is an m n linear system with augmentedmatrix A# . If rank(A# ) rank(A) n then the system hasan infinite number of solutions. Such a system will haven rank(A) free variables.

Solving LinearSystemsSolving linear systems with free variablesMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionExampleUse Gaussian elimination to solvex1 2x2 2x3 x4 3,3x1 6x2 x3 11x4 16,2x1 4x2 x3 4x4 9.Reducing to row-echelon form yieldsx1 2x2 2x3 x4 3,x3 2x4 1.Choose as free variables those variables that do not have apivot in their column.In this case, our free variables will be x2 and x4 . The solutionset is the plane{(5 2s 3t, s, 1 2t, t) : s, t R} .

Solving LinearSystemsThe inverse of a square matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionCan we divide by a matrix? What properties should the inversematrix have?DefinitionSuppose A is a square, n n matrix. An inverse matrix for Ais an n n matrix, B, such thatAB Inand BA In .If A has such an inverse then we say that it is invertible ornonsingular. Otherwise, we say that A is singular.RemarkNot every matrix is invertible.If you have a linear system Ax b and B is an inverse matrixfor A then the linear system has the unique solutionx Bb.

Solving LinearSystemsThe inverse of a square matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionExampleIf 1 1 20 131 A 1A 2 3 3 and B 1 11 1 110 1then B is the inverse of A.Theorem (Matrix inverses are well-defined)Suppose A is an n n matrix. If B and C are two inverses ofA then B C.Thus, we can write A 1 for the inverse of A with no ambiguity.Useful Example a bIf A and ad bc 6 0 then A 1 c d1ad bc d b. c a

Solving LinearSystemsFinding the inverse of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionInverse matrices sound great! How do I find one? Suppose A is a 3 3 invertible matrix. If A 1 x1 x2 x3then 100Ax1 0 , Ax2 1 , and Ax3 0 .001We can find A 1 by solving 3 linear systems at once!In general, form the augmented matrix and reduce to RREF.You end up with A 1 on the right. A In 1In A

Solving LinearSystemsFinding the inverse of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionExample 1 1 2Let’s find the inverse of A 2 3 3 .1 1 1Take the augmented matrix and row reduce. 1 1 2 1 0 0100 2 3 3 0 1 0 0101 1 1 0 0 1001 0 131 11 10 1 {z}A 1Steps1. A12 ( 2)5. A32 ( 1)2. A13 ( 1)6. A31 ( 2)3. M2 ( 1)7. A21 (1)4. M3 ( 1)

Solving LinearSystemsFinding the inverse of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionIn order to find the inverse of a matrix, A, we row reduced anaugmented matrix with A on the left. What if we don’t end upwith In on the left?TheoremAn n n matrix, A, is invertible if and only if rank(A) n.ExampleFind the inverse of the matrix A 1 3.2 6Try to reduce the matrix to RREF. 1 3 A12 ( 2) 1 3 2 60 0Since rank(A) 2, we conclude that A is not invertible.Notice that (1)(6) (3)(2) 0.

Solving LinearSystemsFinding the inverse of a matrixMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionDiagonal matrices have simple inverses.PropositionThe inverse of a diagonalreciprocal entries. a11 . .0matrix is the diagonal matrix with 10 ann 1a11 0 .0 a 1nnUpper and lower triangular matrices have inverses of the sameform.PropositionThe inverse of an upper triangular matrix is upper triangular.The inverse of a lower triangular matrix is lower triangular.

Solving LinearSystemsProperties of inverse matricesMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionSuppose A and B are n n invertible matrices. 1I A 1 is invertible and A 1 A.IIAB is invertible and (AB) 1 B 1 A 1 . 1 TAT is invertible and AT A 1 .CorollarySuppose A1 , A2 , . . . , Ak are invertible n n matrices. Thentheir product, A1 A2 · · · Ak is invertible, and 1 1(A1 A2 · · · Ak ) 1 A 1k Ak 1 · · · A1 .

Solving LinearSystemsUsing inverse matricesMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionRecall that if A is an invertible matrix then the linear systemAx b has the unique solution x A 1 b.ExampleSolve the linear systemx1 3x22x1 5x2 1The coefficient matrix is A 2 1, 3. 3 53 1, so A .52 1The inverse of a 2 2 matrix is 1 1a bd b when ad bc 6 0.c dad bc c a 4x1 53 1 .Hence, 1x22 1 3

Solving LinearSystemsConclusionMath 240Solving ricesDefinitionComputinginversesProperties ofinversesUsing inversematricesConclusionInverse matrices are an elegant way of solving linear systems.They do have some drawbacks:IThey are only applicable when the coefficient matrix issquare.IEven in the case of a square matrix, an inverse may notexist.IThey are hard to compute, at least as complicated asdoing Gauss-Jordan elimination.However, they can be useful ifIthe coefficient matrix has an obvious inverse,Iyou need to solve multiple linear systems with the samecoefficients.

If you have a linear system Ax b and B is an inverse matrix for A then the linear system has the unique solution x Bb: Solving Linear Systems Math 240 Solving Linear Systems Gauss-Jordan elimination . Solve the linear system x 1 3 2 1; 2x 1 5x 2 3: The coe cient matrix is A 1 3 2 5 , so