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Matrices - solving two simultaneous equations

sigma-matrices8-2009-1

One of the most important applications of matrices is to the solution of linear simultaneous equations.

On this leaflet we explain how this can be done.

Writing simultaneous equations in matrix form

Consider the simultaneous equations

x+ 2y= 4

3x-5y= 1

Provided you understand how matrices are multiplied together you will realise that these can be written in matrix form as ?1 23-5?? x y? =?41?

Writing

A=?1 23-5?

, X=?x y? ,andB=?41? we have AX=B This is thematrix formof the simultaneous equations. Here the only unknown is the matrixX, sinceAandBare already known.Ais called thematrix of coefficients.

Solving the simultaneous equations

Given AX=B we can multiply both sides by the inverse ofA, provided this exists, to give A -1AX=A-1B ButA-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

ifAX=B,thenX=A-1B This result gives us a method for solving simultaneous equations. All we need do is write them in matrix form, calculate the inverse of the matrix of coefficients, and finally perform a matrix multiplication. www.mathcentre.ac.uk 1 c?mathcentre 2009

Example.Solve the simultaneous equations

x+ 2y= 4

3x-5y= 1

Solution.We have already seen these equations in matrix form:?1 23-5? ? x y? =?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 by

X=A-1B=-1

11? -5-2 -3 1?? 4 1? =-1 11? -22 -11? ?21? Hencex= 2,y= 1is the solution of the simultaneous equations.

Example.Solve the simultaneous equations

2x+ 4y= 2

-3x+y= 11

Solution.In matrix form:?2 4

-3 1?? x y? =?2 11?

We need to calculate the inverse ofA=?2 4

-3 1? A -1=1 (2)(1)-(4)(-3)? 1-4 3 2? =114? 1-4 3 2?

ThenXis given by

X=A-1B=1

14? 1-4 3 2?? 2 11? 1 14? -42 28?
?-3 2? Hencex=-3,y= 2is the solution of the simultaneous equations. You should check the solution by substitutingx=-3andy= 2into both given equations, and verifying in each case that the left-hand side is equal to the right-hand side. Note that a video tutorial covering the content of this leaflet is available fromsigma. www.mathcentre.ac.uk 2 c?mathcentre 2009quotesdbs_dbs20.pdfusesText_26