[PDF] Matrix inversion of a 3x3 matrix - Mathcentre PDF sigma-matrices11-2009-1.pdf

The adjoint and inverse of a matrix In this leaflet we consider how to find the inverse of a 3×3 matrix Before you work through this leaflet, you will need to know

[PDF] Inverse of a 3x3 Matrix

A method for finding the inverse of a matrix is described in this document The matrix will be used to illustrate the method 1 Matrix of Minors If we go through each

[PDF] 25 Inverse Matrices - MIT Mathematics

To invert a 3 by 3 matrix A, we have to solve three systems of equations: Ax1 D e1 0 3 3 1 1 0 0 1 3 7 7 5 : The next stage creates zeros below the second

[PDF] Inverse of a Matrix using Minors, Cofactors and Adjugate

We can calculate the Inverse of a Matrix by: Step 1: calculating the Matrix of Minors, • Step 2: then turn that into the Matrix of Cofactors, • Step 3: then the Adjugate, and • Step 4: multiply that by 1/Determinant

[PDF] A New Method to Compute the Adjoint and Inverse of a 3 × 3 non

3 non – singular matrices is considered In this method to find the determinant value, adjoint of matrix is very quick when comparing to other known method This is

[PDF] Lec 17: Inverse of a matrix and Cramers rule We are aware of

[Don't confuse with cofactors Aij] Example Solve the linear system 3x1 + x2 − 2x3 = 4 −x1 + 2x2 + 3x3 = 1 2x1

[PDF] The Inverse of a Matrix - Learn

We develop a rule for finding the inverse of a 2 × 2 matrix (where it exists) and Suppose we wish to find the inverse of the matrix A = 1 3 3 1 4 3 2 7 7

[PDF] 55 The inverse of a matrix

matrix is zero, then it will not have an inverse, and the matrix is said to be singular Only non-singular matrices have inverses 2 A formula for finding the inverse

[PDF] The Inverse Matrix

8 Inverse Matrix In this section of we will examine two methods of finding the inverse of a matrix, these are • The adjoint method • Gaussian Elimination

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

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)))

quotesdbs_dbs7.pdfusesText_5

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

Matrix inversion of a3×3matrix

The adjoint and inverse of a matrix

A=(((7 2 10 3-1

C=(((-2 3 9

8-11-34

3-11 7

9-34 21)))

Then the formula for the inverse matrix is

Given a matrixA, its inverse is given by

The inverse has the special property that

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

Solution

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

3-11 7

9-34 21)))

3-11 7

9-34 21)))

3-11 7

9-34 21)))

Solving a set of simultaneous equations

7x+ 2y+z= 21

3y-z= 5

In matrix form these equations can be written

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

3-11 7

9-34 21)))