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5.6 Using the inverse matrix to solve equations

The inverse matrix tosolve equationsIntroductionOne of the most important applications of matrices is to the solution of linear simultaneousequations. On this leaflet we explain how this can be Writing simultaneous equations in matrix formConsider the simultaneous equationsx+ 2y= 43x 5y= 1 Provided you understand how matrices are multiplied together you will realise that these canbe written in matrix form as(1 23 5)(xy)=(41)WritingA=(1 23 5),X=(xy),andB=(41)we haveAX=BThis is thematrix formof the simultaneous equations . Here the unknown is the matrixX,sinceAandBare already called thematrix of solving the simultaneous equationsGivenAX=Bwe can multiply both sides by the inverse ofA, provided this exists, to giveA 1AX=A 1 BButA 1A=I, the identity matrix . Furthermore,IX=X, because multiplying any matrix byan identity matrix of the appropriate size leaves the matrixunaltered. SoX=A Pearson Education Ltd2000ifAX=B,thenX=A 1 BThis result gives us a method for solving simultaneous equations .

2. Solving the simultaneous equations Given AX = B we can multiply both sides by the inverse of A, provided this exists, to give A−1AX = A−1B But A−1A = I, the identity matrix. Furthermore, IX = X, because multiplying any matrix by an identity matrix of the appropriate size leaves the matrix unaltered. So X = A−1B www.mathcentre.ac.uk 5.6.1

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Transcription of 5.6 Using the inverse matrix to solve equations

1 The inverse matrix tosolve equationsIntroductionOne of the most important applications of matrices is to the solution of linear simultaneousequations. On this leaflet we explain how this can be Writing simultaneous equations in matrix formConsider the simultaneous equationsx+ 2y= 43x 5y= 1 Provided you understand how matrices are multiplied together you will realise that these canbe written in matrix form as(1 23 5)(xy)=(41)WritingA=(1 23 5),X=(xy),andB=(41)we haveAX=BThis is thematrix formof the simultaneous equations . Here the unknown is the matrixX,sinceAandBare already called thematrix of solving the simultaneous equationsGivenAX=Bwe can multiply both sides by the inverse ofA, provided this exists, to giveA 1AX=A 1 BButA 1A=I, the identity matrix . Furthermore,IX=X, because multiplying any matrix byan identity matrix of the appropriate size leaves the matrixunaltered. SoX=A Pearson Education Ltd2000ifAX=B,thenX=A 1 BThis result gives us a method for solving simultaneous equations .

2 All we need do is write themin matrix form, calculate the inverse of the matrix of coefficients, and finally perform a the simultaneous equationsx+ 2y= 43x 5y= 1 SolutionWe have already seen these equations in matrix form:(1 23 5)(xy)=(41)We need to calculate the inverse ofA=(1 23 5).A 1=1(1)( 5) (2)(3)( 5 2 3 1)= 111( 5 2 3 1)ThenXis given byX=A 1B= 111( 5 2 3 1)(41)= 111( 22 11)=(21)Hencex= 2,y= 1 is the solution of the simultaneous solve the following sets of simultaneous equations usingthe inverse matrix )5x+y= 133x+ 2y= 5b)3x+ 2y= 2x+ 4y= 6 Answers1. a)x= 3, y= 2, b)x= 2, y= 2 . Pearson Education Ltd2000


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