The inverse of an invertible matrix is denoted A-1 Also, when a matrix is invertible, so is its inverse, and its inverse's inverse is itself, (A-1)-1 = A If A and B are both invertible, then their product is, too, and (AB)-1 = B-1A-1
| Previous PDF | Next PDF |
[PDF] 25 Inverse Matrices - MIT Mathematics
It fails to have two pivots as required by Note 1 Elimination turns the second row of this matrix A into a zero row The Inverse of a Product AB
[PDF] 10 Inverse Matrix Three Properties of the Inverse - UC Davis
M = I = M −1 M Inverse of a 2 × 2 Matrix Let M and N be the matrices: M = ( a b c d \ , N = ( d −b −c a \ Multiplying these matrices gives: MN = ( ad − bc 0
[PDF] Matrices, transposes, and inverses
1 fév 2012 · This is a requirement in order for matrix multiplication to be defined x A The notion of an inverse matrix only applies to square matrices
[PDF] Matrix Inverse: A Properties Inverse of an Inverse Inverse of a Matrix
Matrix Inverse: A –1 An n × n matrix A is said to have the inverse A–1 if the following two commutative matrix product relations are satisfied: A A–1 = I and
[PDF] The inverse of a 2x2 matrix - Mathcentre
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 1 1 ) and
[PDF] Lecture 6 Inverse of Matrix
Any elementary row operation is equivalent to left multiplying by the corresponding elementary matrix ² Justification of LU Decomposition Algorithm Recall in
[PDF] Invertible matrix
24 jan 2013 · The matrix A can be expressed as a finite product of elementary matrices The matrix A has a left inverse (i e there exists a B such that BA = I)
[PDF] Matrix multiplication Diagonal matrices Inverse matrix - TAMU Math
Diagonal matrices Inverse matrix Scalar multiplication: to multiply a matrix A by a scalar r, one The product of matrices A and B is defined if the number of
[PDF] inverse relationship science
[PDF] inverseur de source courant continu
[PDF] inverter layout
[PDF] invertible linear transformation
[PDF] invest in 7 eleven
[PDF] investigatory project in physics for class 12 cbse pdf
[PDF] investing in hilton hotels
[PDF] investment grade rating
[PDF] investor pitch presentation example
[PDF] investor presentation (pdf)
[PDF] investor presentation ppt template
[PDF] invité politique dimanche france inter
[PDF] invité politique matinale france inter
[PDF] invoice declaration
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