Sec 3.5 Inverses of Matrices

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Sec 3.5 Inverses of Matrices
Where A is nxn
Finding the inverse of A
Seq or row operations
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Finding the inverse of A
A
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Properties
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Fact1 AB in terms of columns of B
Fact1 Ax in terms of columns of A
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Basic unit vector
J-th location
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What is The Big Day
Register ???
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Def A is invertable if
There exists a matrix B such that
TH1 the invers is unique
TH2 the invers of 2x2 matrix
Find inverse
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TH3 Algebra of inverse If A and B are
invertible, then
1
2
3
4
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TH4 solution of Ax b
Solve
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Def E is elementary matrix if
1) Square matrix nxn
2) Obtained from I by a single row operation
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REMARK
Let E corresponds to a certain elem row
operation. It turns out that if we perform this
same operation on matrix A , we get the product
matrix EA
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NOTE
Every elementary matrix is invertible
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Sec 3.5 Inverses of Matrices
TH6
A is invertible if and only if it is row
equivalent to identity matrix I
Row operation 1
Row operation 2
Row operation 3
Row operation k
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Solving linear system
Solve
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Matrix Equation
Solve
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Definition
A is nonsingular matrix if the system
has only
the trivial solution
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TH7
A is an nxn matrix. The following is
equivalent
(a) A is invertible
(b) A is row equivalent to the nxn identity
matrix I
(c) Ax 0 has the trivial solution
(d) For every n-vector b, the system A x b
has a unique solution
(e) For every n-vector b, the system A x b
is consistent