5.6 Using the inverse matrix to solve equations
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
Solution by Inverse Matrix Method
Matrix algebra allows us to write the solution of the system using the inverse matrix of the coefficients. In practice the method is suitable only for small
PRECIPITATION REACTIONS
∴x = 2. Matrices equations can also be solved by using an inverse matrix. Example 3. Solve the following matrix equation: . 3 4. −1 2.
Adjustment of an Inverse Matrix Corresponding to a Change in One
The utility of a computational method for obtaining the inverse of a matrix would be increased considerably if the inverse could be transformed in a simple.
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ALGORITHM AND INVERSE MATRIX METHOD FOR STEADY-. STATE ANALYSIS OF WATER DISTRIBUTION NETWORKS. Selami DEMİR (ORCID: 0000-0002-8672-9817)*. Çevre Mühendisliği
A Simple Method for Computing the Inverse of a Numerator
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Computation of the Inverse of a Matrix
Their method is discussed in detail in Elementary Matrices by Frazer Duncan & Collar. It is a special case of the method to be presented in this paper. Let us
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Cornell University. 1. Summary. The enlargement principle provides techniques for inverting any nonsingular matrix by building the inverse upon the inverses
5.6 Using the inverse matrix to solve equations
All we need do is write them in matrix form calculate the inverse of the matrix of coefficients
5.6 Using the inverse matrix to solve equations
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
2.5 Inverse Matrices
Note 6 A diagonal matrix has an inverse provided no diagonal entries are zero: Now multiply F by the matrix E in Example 2 to find FE. Also multiply E.
A Simple Method for Computing the Inverse of a Numerator
The inverse of a numerator relationship matrix is needed for best linear unbiased prediction of breeding values. The purpose of this paper to is present a
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methods the amount of computation increases rapidly with increase in order of the matrix. The utility of a computational method for obtaining the inverse
A Simple Method for Computing the Inverse of a Numerator
The inverse of a numerator relationship matrix is needed for best linear unbiased prediction of breeding values. The purpose of this paper to is present a
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The inverse of a 2 × 2 matrix
an inverse matrix and how the inverse of a 2 × 2 matrix is calculated. Preliminary example. Suppose we calculate the product of the two matrices ( 4 3.
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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 how to find the determinant and
[PDF] 55 The inverse of a matrix - Mathcentre
The inverse of a square n × n matrix A is another n × n matrix denoted by A?1 such that AA?1 = A?1A = I where I is the n × n identity matrix
[PDF] 8 Inverse Matrix
Another useful method used to find an inverse of matrix involves subjecting our matrix to a series of elementary row operations 8 2 1 Operation: Elementary Row
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The following examples will show a method to solve for the inverse of a matrix Example 1: Find the inverse of the matrix
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We will illustrate this by finding the inverse of a 3 × 3 matrix First of all we need to define what it means to say a matrix is in reduced row echelon form
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A matrix A that has an inverse is called an invertible matrix 8 Example 2 4 1 The argument in Example 2 4 2 shows that no zero matrix has an inverse
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Here we give a method for finding the inverse of a square matrix We will see that this involves nothing more than row reduction that we have seen before For
How to do the inverse matrix method?
To find the inverse of a 2x2 matrix: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).What is inverse method method?
In modeling and simulation, the inverse method consists in a technique where model input parameters are estimated (with uncertainty) from comparison of model output magnitudes with experimental data.How do you Find the Inverse of the 3 by 3 Matrix?
1Estimate the determinant of the given matrix.2Find the transpose of the given matrix.3Calculate the determinant of the 2 x 2 matrix.4Prepare the matrix of cofactors.5At the last, divide each term of the adjugate matrix by the determinant.
8 Inverse Matrix
In this section of we will examine two methods of finding the inverse of a matrix, these areThe adjoint method.
Gaussian Elimination.
8.1 Matrix Inverse: The Adjoint Method
We require a couple of definitions before we set out the procedure to find the inverse of a matrix.8.1.1 Type of Matrix: Cofactor matrix
Definition 8.1(Cofactor Matrix).
Given an×nmatrixA. The cofactor matrixCofAis the matrix formed by evaluating the cofactors of each entry inA C=( (C11C12... C1n
C21C22... C2n.
C n1Cn2... Cnn) Example 8.1.1(Cofactor Matrix).Find the cofactor matrix for A=( (-1 1 0 2 0 01 1-2)
Solution:
In order to find the cofactor matrix forAwe will need the cofactors of each and every entry inA, C11= (-1)1+1M11= (-1)2det?0 0
1-2? = 0 C12= (-1)1+2M12= (-1)3det?2 0
1-2? = 4 C13= (-1)1+3M13= (-1)4det?2 01 1?
= 2 41continuing this process (you should check this) we will find C
21= 2, C22= 2, C23= 2
C31= 0, C32= 0, C33=-2
and thus the cofactor matrix is C=( (0 4 22 2 20 0-2)8.1.2 Adjoint of a matrix
Definition 8.2(Adjoint of a matrix).
The adjoint of a matrixAdenoted adj(A) is simply thetranspose of the of the cofactor matrix. That is, ifCdenotes the cofactor matrix ofAthen adj(A) =C?Example 8.1.2(The adjoint).
Find the adjoint of the matrix
A=( (-1 1 0 2 0 01 1-2)
Solution:
We have done all the hard work (finding the cofactor matrix) inthe previous example C=( (0 4 22 2 20 0-2) thus, adj(A) =C?=( (0 2 04 2 02 2-2) 428.1.3 The Inverse: Using the adjointWe are now ready to state (without proof) a useful theorem which will allow us to com-
pute the inverse of a matrix.Theorem 8.1.1(Inverse using the adjoint).
LetAbe an×nmatrix. If detA?= 0, then
A -1=1 detAadjA The steps involved in finding an inverse using an the adjoint method for a matrixA1. Find the determinant of the matrix of interest detA
If detA?= 0 then the inverse will exist.
If detA= 0 or matrix isn"t square then the inverse will not exist.2. Find the cofactor matrixC, by finding the cofactor for each element ofA.
The cofactor of theith-rowjth-column element ofAis C ij= (-1)i+jMij whereMijis the minor.3. Find the adjoint ofA
adjA=C?4. The inverse is given by
A -1=1 detAadjAExample 8.1.3(The Inverse).
Find the inverse of
A=( (-1 1 0 2 0 01 1-2)
using the adjoint method.Solution:
43We have the cofactor matrix and the adjoint ofA
C=( (0 4 22 2 20 0-2) )and adj(A) =( (0 2 04 2 02 2-2) We can find the determinant ofAby performing a cofactor expansion about any row or column ofA. Picking the third column (as it has two zeros) we have detA=a13C13+a23C23+a33C33 we have all the cofactors (from the cofactor matrix) thus, detA= (0)(2) + (0)(2) + (-2)(-2) = 4. According to our theorem concerning the adjoint and the inverse of a matrix we have A -1=1 detAadjA=14( (0 2 04 2 02 2-2) and thus, A -1=( (0 1/20 1 1/201/21/2-1/2)
We can check if this is in fact the inverse
AA -1=( (-1 1 0 2 0 01 1-2)
(0 1/20 1 1/201/21/2-1/2)
(1 0 00 1 00 0 1) and A -1A=( (0 1/20 1 1/201/21/2-1/2)
(-1 1 0 2 0 01 1-2)
(1 0 00 1 00 0 1) and thus we have an inverse. 448.2 Matrix Inverse: Gaussian Elimination MethodAnother useful method used to find an inverse of matrix involvessubjecting our matrix
to a series ofelementary row operations.8.2.1 Operation: Elementary Row Operations
There are three types of elementary tow operations1. Add/subtract a multiple of one row to another row.
2. Multiply a row by a constant.
3. Interchange two rows.
Interestingly these elementary row operations have very specific effects on the determi- nant of a matrix.Row OperationEffect on determinant
Add a multiple of one row to another rowNone
Multiply a row by a constantkmultiplied byk
Interchange two rowsmultiplied by-1.
How can this be used to find a determinant for matrix? We can reduce a matrixA to upper triangular form using elementary row operations making it a matrixA?. The determinant ofA?is easy to find (as it is triangular the determinant is simply the product of the entries on the diagonal) and relate its determinant to the determinant ofAby working back through the row operations that were used in thereduction process. Example 8.2.1(The determinant using elementary row operations).Find the determinant of
A=( (2 4 91 2 41 10 7) using elementary row operations.Solution:
(2 4 91 2 41 10 7) )R3 to R3-R2---------→(det unchanged)( (2 4 91 2 40 8 3) )swap R2 and R3---------→(det× -1)( (2 4 90 8 31 2 4) )2×R3-----→(det×2)( (2 4 90 8 32 4 8)R3 to R3-R1---------→(det unchanged)(
(2 4 90 8 30 0-1) 45We now have the matrixAtransformed into an upper triangular matrix A (2 4 90 8 30 0-1) the determinant ofA" is given by the product of the elements on the diagonal detA?= (2)(8)(-1) =-16 The operations that we conducted on the matrixAwere
Row OperationEffect on determinant
Add a multiple of one row to another rowdet unchangedInterchange two rowsmultiplied det by-1
Multiply a row by 2multiplied det by 2
Add a multiple of one row to another rowdet unchanged and thus, detA?= (-1)(2)detA thus, detA= 8.8.2.2 Matrix inverse using row operations
We can use these row operations to find the inverse of a matrix, the result that we will use is quoted here without proof. If a sequence of elementary row operations on a square matrixAcan reduce the matrix to the identity matrixI, then the same sequence of row operations applied toIwill result inIbeing transformed toA-1.Of note is that
If it"s not possible to reduceAtoIusing elementary row operations thenAis not invertible. IfAis invertible then there will be more than one way to reduce ittoI. Since we are going to perform the same operations on a given matrix A=( (a11a12a13
a21a22a23
a31a32a33)
)andI=( (1 0 00 1 00 0 1) 46We will introduce the followingaugmented matrix, which will allow us to manipulate both matrices at the same time easily (a
11a12a13
1 0 0 a21a22a23
0 1 0 a31a32a33
0 0 1)
which is nothing more than both the matrices placed adjacentto one another. Example 8.2.2(Inverse using row operations and an augmented matrix).Find the inverse of
A=( (0 1 21 2 02 0 1) using elementary row operations.Solution:
Step 1:Augment the matrix with the identity matrix (0 1 2 1 0 0 1 2 0 0 1 0 2 0 10 0 1)
Step 2:Swap rows (and multiply by a constant if necessary) to ensurethat the left side of the augmented matrix will have a "1" in the first row first column entry (0 1 2quotesdbs_dbs20.pdfusesText_26[PDF] inverse of a 3x3 matrix worksheet
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