Note on the Generalized Inverse of a Matrix Product
NOTE ON THE GENERALIZED INVERSE OF A MATRIX PRODUCT*. T. N. E. GREVILLEt. It is well known that the Moore-Penrose generalized inverse of a matrix.
Matrix inversion Math 130 Linear Algebra
and that A is an inverse of B. If a matrix has no The inverse of an invertible matrix is ... follows from associativity of matrix multiplication.
Note on the Generalized Inverse of a Matrix Product
NOTE ON THE GENERALIZED INVERSE OF A MATRIX PRODUCT*. T. N. E. GREVILLE. It is well known that the Moore-Penrose generalized inverse of a matrix.
2.5 Inverse Matrices
might not exist. What a matrix mostly does is to multiply a vector x. Multiplying Ax D b by A. 1 gives
The Matrix Cookbook
Nov 15 2012 determinant
A Hyperpower Iterative Method for Computing Matrix Products
MATRIX PRODUCTS INVOLVING THE GENERALIZED INVERSE*. JAMES M. GARNETT III? puting the matrix product AtB or BAt
Appendix A: Summary of Vector/Matrix Operations
Vector - matrix multiplication is defined as for matrix - matrix The inverse of the product of two matrices is the reversed product of the inverses:.
Some Mixed-Type Reverse Order Laws for the Moore-Penrose
Nov 7 2003 Key Words and phrases. Mixed-type reverse-order law
The Forward Order Law for Least Squareg-Inverse of Multiple Matrix
Mar 19 2019 Keywords: forward order law; generalized inverse; maximal rank; matrix product; generalized Schur complement. 1. Introduction.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal
Diagonal matrices. Inverse matrix. Scalar multiplication: to multiply a matrix A by ... The product of matrices A and B is defined if the.
[PDF] The inverse of a 2 × 2 matrix - Mathcentre
In this leaflet we explain what is meant by an inverse matrix and how the inverse of a 2 × 2 matrix is calculated Preliminary example Suppose we calculate the
[PDF] Matrices transposes and inverses
1 fév 2012 · This is a requirement in order for matrix multiplication to be defined The notion of an inverse matrix only applies to square matrices
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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 1 4 1 3 A ? ?
[PDF] 24 Matrix Inverses - Mathemoryedu
The idea is the Inverse Criterion: If a matrix B can be found such that AB = I = BA then A is invertible and A?1 = B Example 2 4 8 If A is an invertible
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The product of the two matrices is indeed the identity matrix so we're done Linear Systems and Inverses If M?1 exists and is known then we can immediately
[PDF] Matrix Operations Section 22: The Inverse of a Matrix
We can add two matrices of the same size just by adding their components We can multiply a matrix by a scalar just by multiplying each entry by that
[PDF] Fast Matrix Multiplication and Inversion Notes for Math 242 Linear
We will also show somewhat surprisingly that one can also compute the inverse of a matrix with a number of compu- tations that is not markedly different from
[PDF] Matrix inversion Math 130 Linear Algebra
and that A is an inverse of B If a matrix has no The inverse of an invertible matrix is follows from associativity of matrix multiplication
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Inverse for matrix product • A and B are invertible nxn matrices is AB invertible? • Let 1 2 ? be nxn invertible matrices The product
[PDF] Lecture 6 Inverse of Matrix
Left-multipling the matrix equation by the inverse matrix C = A-1 we have Multiplying the above equation by B from the left we find B (BC) = BI2
What is the inverse of a matrix product?
5. inverse of a matrix product: inv(A * B) = inv(B) %*% inv(A) This extends to any number of terms: the inverse of a product is the product of the inverses in reverse order.What is inverse product?
Let A,B?Fn×n where F denotes a field and n is a positive integer. Let C=AB. If A and B are both invertible, then C is invertible and C?1 is given by B?1A?1. (How to find the inverse of a matrix as a product of elementary matrices?
In fact, the inverse of an elementary matrix is constructed by doing the reverse row operation on I. E?1 will be obtained by performing the row operation which would carry E back to I. If E is obtained by switching rows i and j, then E?1 is also obtained by switching rows i and j.- What is the Inverse of 3x3 Matrix? The inverse of a 3x3 matrix, say A, is a matrix of the same order denoted by A-1 where AA-1 = A-1A = I, where I is the identity matrix of order 3x3. i.e., I = ???100010010??? [ 1 0 0 0 1 0 0 1 0 ] .
Matrix inversion
Math 130 Linear Algebra
D Joyce, Fall 2015
We'll start o with the denition of the inverse
of a square matrix and a couple of theorems.Denition 1.We say that two squarennma-
tricesAandBareinversesof each other ifAB=BA=I
and in that case we say thatBis an inverse ofA and thatAis an inverse ofB. If a matrix has no inverse, it is said to besingular, but if it does have an inverse, it is said to beinvertibleornonsingular.Theorem 2.A matrixAcan have at most one
inverse. The inverse of an invertible matrix is denotedA1. Also, when a matrix is invertible, so is its inverse, and its inverse's inverse is itself, (A1)1=A.Proof.Suppose thatBandCare both inverses of
A. Then bothAB=BA=IandAC=CA=I.
Therefore
B=BI=B(AC) = (BA)C=IC=C
Thus, there is at most one inverse.
The second statement (A1)1=Afollows from
the denition of the inverse ofA1, namely, its in- verse is the matrixBsuch thatA1B=BA1=I.SinceAhas that property, thereforeAis the inverse
ofA1.q.e.d.Theorem 3.IfAandBare both invertible, then
their product is, too, and (AB)1=B1A1.Proof.Since there is at most one inverse ofAB,
all we have to show is thatB1A1has the prop- erty required to be an inverse ofAB, name, that (AB)(B1A1) = (B1A1)(AB) =I. But that follows from associativity of matrix multiplication and the facts thatAA1=A1A=IandBB1=1B=I.q.e.d.Inverses of22matrices.You can easily nd
the inverse of a 22 matrix. Consider a generic22 matrix
A=a b c dIt's inverse is the matrix
1=d=b=
c=a= where is the determinant ofA, namely =adbc; provided is not 0. In words, to nd the inverse of a 22 matrix, (1) exchange the entries on the major diagonal, (2) negate the entries on the mi- nor diagonal, and (3) divide all four entries by the determinant.It's easy to verify thatA1actually is the inverse
ofA, just multiply them together to get the identity matrixI.A method for nding inverse matrices.Next
we'll look at a dierent method to determine if an nnsquare matrixAis invertible, and if it is what it's inverse is.The method is this. First, adjoin the identity
matrix to its right to get ann2nmatrix [AjI]. Next, convert that matrix to reduced echelon form. If the result looks like [IjB], thenBis the desired inverseA1. But if the square matrix in the left half of the reduced echelon form is not the identity, thenAhas no inverse.We'll verify that this method works later.
Example 4.Let's illustrate the method with a 3
3 example. LetAbe the matrix
A=2 432 41 0 2
0 1 03
Form the 36 matrix [AjI], and row reduce it.
I'll use the symbolwhen a row-operation is ap-
plied. Here are the steps. [AjI] =2432 41 0 0
1 0 20 1 0
0 1 00 0 1
41 0 20 1 0
0 1 00 0 1
32 41 0 0
41 0 20 1 0
0 1 00 0 1
02213 03
41 0 20 1 0
0 1 00 0 1
0 0213 23
41 0 012 2
0 1 00 0 1
0 0 11=2 3=213
= [IjA1]This row-reduction to reduced echelon form suc-
ceeded in turning the left half of the matrix into the identity matrix. When that happens, the right half of the matrix will be the inverse matrixA1.Therefore, the inverse matrix is
1=2 412 20 0 1
1=2 3=213
Matlabcan compute inversesor tell you if
they're singular. >> A = [1 2; 3 4] A = 1 2 3 4 >> B = inv(A) B = -2.0000 1.00001.5000 -0.5000
>> A*B ans =1.0000 00.0000 1.0000
>> C = [1 2; 3 6] C = 1 2 3 6 >> D = inv(C)Warning: Matrix is singular to working precision.
D =Inf Inf
Inf Inf
Math 130 Home Page at
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