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1 is called “A inverse ” DEFINITION The matrix A is invertible if there exists a matrix A 1 such that A 1 A D I and AA 1 D I: (1) Not all matrices have inverses

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[PDF] Matrix inversion of a 3x3 matrix - Mathcentre

In this leaflet we consider how to find the inverse of a 3×3 matrix Before you Example Find the inverse of A = ⎛ ⎢ ⎝ 7 2 1 0 3 −1 −3 4 −2 ⎞ ⎢ ⎠

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Matrix inversion of a3×3matrix

sigma-matrices11-2009-1

The adjoint and inverse of a matrix

In this leaflet we consider how to find the inverse of a3×3matrix. Before you work through this leaflet,

you will need to know how to find thedeterminantandcofactorsof a3×3matrix. If necessary you should refer to previous leaflets in this series which cover these topics. Here is the matrixAthat we saw in the leaflet on finding cofactors and determinants. Alongside, we have assembled the matrix of cofactors ofA.

A=(((7 2 10 3-1

-3 4-2)))

C=(((-2 3 9

8-11-34

-5 7 21)))

In order to find the inverse ofA, we first need to use the matrix of cofactors,C, to create theadjoint

of matrixA. The adjoint ofA, denoted adj(A), is the transpose of the matrix of cofactors: adj(A) =CT Remember that to find the transpose, the rows and columns are interchanged, so that adj(A) =CT=(((-2 8-5

3-11 7

9-34 21)))

Then the formula for the inverse matrix is

A -1=1 det(A)adj(A) where det(A)is the determinant ofA.

Given a matrixA, its inverse is given by

A -1=1 det(A)adj(A) where det(A)is the determinant ofA, and adj(A)is the adjoint ofA.

The inverse has the special property that

AA -1=A-1A=I(an identity matrix) www.mathcentre.ac.uk 1 c?mathcentre 2009

ExampleFind the inverse ofA=(((7 2 10 3-1

-3 4-2)))

Solution

We already have thatadj(A) =(((-2 8-5

3-11 7

9-34 21)))

In an earlier leaflet, the determinant of this matrixAwas found to be 1. So A -1=1 det(A)adj(A) =11((( -2 8-5

3-11 7

9-34 21)))

=(((-2 8-5

3-11 7

9-34 21)))

You should verify this is correct by showing thatAA-1=A-1A=I, the3×3identity matrix.

Solving a set of simultaneous equations

We now show how the inverse is used to solve the simultaneous equations:

7x+ 2y+z= 21

3y-z= 5

-3x+ 4y-2z=-1

In matrix form these equations can be written

(7 2 10 3-1 -3 4-2))) (x y x))) =(((21 5 -1)))

Recall that whenAX=B, thenX=A-1Bso

(x y x))) =(((-2 8-5

3-11 7

9-34 21)))

(21 5 -1))) =(((-42 + 40 + 5

63-55-7

189-170-21)))

=(((31 -2)))

Sox= 3,y= 1andz=-2.

These values should be checked by substituting them back into the original equations.

Finally, note that if the determinant of the coefficient matrixAis zero, then it will be impossible to

find the inverse ofA, and this method will not be applicable. 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