Review of Matrices Or A Fast Introduction
Fundamental Tools For Linear Systems Why Study Matrices? Fundamental Tools For Linear Systems
Matrices Come From Systems of Equations We’ve known how to find For different right hand sides
Matrices Come From Systems of Equations Matrix Vector Matrices are all about what happens with different
Matrices Come From Systems of Equations Matrices are all about what happens with different
Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication (Think of it as a system of equations!) Try it!
Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication (Think of it as a system of equations!) Try it!
Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication Rows Notation Columns Second Index is Column First Index Is Row
Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication Notation Second Column First Row
Matrices Come From Systems of Equations Other Important Definitions Addition
Matrices Come From Systems of Equations Other Important Definitions Addition
Matrices Come From Systems of Equations Other Important Definitions Addition
Matrices Come From Systems of Equations Other Important Definitions Addition
Matrices Come From Systems of Equations Other Important Definitions Addition
Matrices Come From Systems of Equations Other Important Definitions Addition
Matrices Come From Systems of Equations Other Important Definitions Addition In General
Matrices Come From Systems of Equations Other Important Definitions Addition In General
Matrices Come From Systems of Equations Other Important Definitions Addition In General
Matrices Come From Systems of Equations Other Important Definitions Scalar Multiplication
Matrices Come From Systems of Equations Other Important Definitions Addition and Scalar Multiplication Give Subtraction Without a Vector
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Matrices Come From Systems of Equations Other Important Definitions Matrix Multiplication
Special Matrices Identity Zero
Matrix Algebra Rules A Lot Like Normal Algebra Rules (Addition Commutes) (Addition Associates) (Multiplication Associates) (Distribute) (Scalars Distribute) Notice: (Multiplication DOES NOT Commute)
Integration and Differentiation
Integration and Differentiation
Matrix Inversion Many Matrices Have an Inverse Inverse Has Special Properties For Example:
Inverse Exists If and Only If Inverse Formula Matrix Inverse For 2 by 2 The Determinant Inverse Exists If and Only If
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
Inverse Helps Solve Systems of Equations
The Inverse Does Not Exist! Singular Matrices Matrix Inverse For 2 by 2 If the Determinant is 0 The Inverse Does Not Exist! May Not Have A Solution
Example Determinant is 0 Impossible!
If (and only if) the Determinant is 0 Really Important Fact If (and only if) the Determinant is 0 There is a Vector Where
Example
Three Ways of Saying The Same Thing (1) The Determinant is 0 (2) The Matrix Has No Inverse (Is Singular) (3) There Is A Vector Where
Summary Matrices Come From Systems Of Equations We can do Algebra on Matrices Matrices are called “singular” if the determinant is 0. Such matrices have no inverses.
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