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More on Inverse
Last Week Review
• Matrix
–
–
–
–
–
Rule of addition
Rule of multiplication
Transpose
Main Diagonal
Dot Product
• Block Multiplication
• Matrix and Linear
Equations
– Basic Solution
• X1 + X0
– Linear Combination
– All solutions of LES
• Inverse
– Det
– Matrix Inversion
Method
• Double matrix
Warm Up
• Find the inverse of
• Using matrix inversion method
– [ A I ]  [ I A-1 ]
Solution
• Start with the double matrix
• Swap 1 with 2
• R2 – 2R1, R3 – R1
• More to Reduced Row Echelon Form
PROPERTIES OF INVERSE
Transpose and Inverse
• If A is invertible, show that
– AT is also invertible
– (AT)-1 = (A-1)T
Solution
• A-1 exists
– Its transpose is the inverse of AT
• So
AT(A-1)T = (A-1A)T = IT = I
(A-1)TAT = (AA-1)T = IT = I
Inverse of Multiplication
• If A and B are invertible, show that
– AB is also invertible
– (AB)-1 = B-1A-1
Solution
• Assume that (AB)-1 exists
– And it is B-1A-1
• (B-1A-1)(AB) = B-1(A-1A)B = B-1IB = B-1B = I
• (AB)(B-1A-1) = A(BB-1)A-1 = AIA-1 = AA-1 = I
• Hence, it is actually the inverse
Rule of Inverse
Inverse Equivalence
• A is invertible
• The homogeneous system AX = 0 has only
the trivial solution X = 0
• A can be carried to the identity matrix In by
elementary row operation
• The system AX=B has at least one solution X
• There exists an n x n matrix C such that AC =
In
ELEMENTARY MATRICES
Elementary Matrix
• A matrix that can be obtained from I by
single elementary row operation
• Example
Elementary Operation
• Interchange two equations
• Multiply one equation with a
number
• Add a multiple of one equation to a
equation
Lemma
• If an elementary row operation is
performed on an
matrix
• The result is
where is the elementary
matrix
is obtained by performing the same operation
on
identity matrix.
Inverse of elementary operation
• Each operation has an inverse
– Also an elementary operation
• So are the elementary matrix
Operation
Inverse
Interchange row p and q
Interchange row q and p
Multiply row p by k != 0
Multiply row p by 1/k
Add k times row p to row q != p
Subtract k times row p to row q
Inverse of Elementary Matrix
• Hence, each elementary matrix E has its
inverse
• The inverse change E back to I
Lemma 2
• Every elementary matrix E is invertible
– Its inverse is also an elementary matrix
• Of the same type as well
• It also corresponds to the inverse of the row
operation that produce E
Inverse and Rank
• Suppose that A  B by a series of
elementary row operation
• Hence
– A  E1A  E2E1A  EkEk-1…E2E1A  B
• i.e., A  UA = B
– Where U = EkEk-1…E2E1
• U is invertible
– Why?
Finding U
• AB by some elementary row operations
• Perform the same operations on I
• Doing the same thing just like the matrix
inversion algorithm
• [A I]  [B U]
Theorem: Property of U
• Suppose that A is m x n and A  B by some
sequence of elementary row operations
– B = UA where U is m x m invertible matrix
– U can be computed by [A I]  [B U] using the
same operations
– U = EkEk-1…E2E1 where each Ei is the elementary
matrix corresponding to the elementary row
operation
U and
-1
A
• Suppose that A is invertible
– We know that A  I
– So, let B be I
– Hence, [A I]  [I U]
• I = UA
• i.e., U = A-1
• This is exactly the matrix inversion algorithm
– But, A-1 =U = EkEk-1…E2E1
– Hence A = (A-1)-1 = (EkEk-1…E2E1)-1
= E1-1E2-1…Ek-1-1Ek-1
• This means that every invertible matrix is a product of
elementary matrices!!!
Theorem 2
• A square matrix is invertible if and only if it
is a product of elementary matrices.
TRANSFORMATION
Ordered n-tuple (Vector)
• Let be the set of real number
• If n >= 1, an ordered sequence
– (a1,a2,..,an) is called an ordered n-tuple
denotes the set of all ordered n-tuples
• The ordered n-tuple is also called vectors
Transformation
• A function T from
• Written T:

to
domain
codomain
• To describe T, we must give the definition of
all T(X) for every X in
• T and S is the same if T(X) = S(X) for every X
– That is the definition of T
Matrix Transformation
• A transformation such that
– T(X) is AX
• Called the matrix transformation induced by
A
– If A = 0, it is called the zero transformation
– If A = I, it is called the identify transformation
Example
• X-expansion
• Induced by
Example
• Reflection
• Induced by
Example
• X-shear
• Induced by
Translation is not Linear Transform
• Translation
– T(X) = X + w
• If it is, then
– X + w = AX for some A
– What if a = 0?
Linear Transformation
• A transformation is called a linear
transformation when
– T(X + Y) = T(X) + T(Y)
– T(aX) = aT(X)
Linear Transform and Matrix Transform
• Let T: RN  RM be a transformation
– T is linear if and only if it is a matrix
transformation
– If T is linear, then T is induced by a unique
matrix A
Composition
• Transform of a transform
• ST = S(T(X))
Composition
• If R,S,T are linear transformation
– Compositions of them are also linear
– Is associative
• (since it is matrix transform)
Inverse through transform
• Inverse of the transform is the inverse of the
function
• Hence, domain and codomain must be the
same
• Given a linear transformation
– It’s inverse is induced by A-1