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# Matrix Properites

### Matrices, Determinents, Methods

Question | Answer |
---|---|

When will a Matix (A) NOT have an inverse? | 0 in diagonal det(A) = 0 2 rows same or multiple (after Gaussina) Row of 0 |

What must be SQUARE? | Matrices with Inverse Determints Upper/Lower Triangular Matrices Diagonal Matrices |

Properties of Determint | det(A T) = det (A) det(AB) = det(A) det(B) BUT det(A+B) NOT = to det (A)+det(B) Mulitply diagonal when Upper/Lower/Diagonal |

det(A -1) = | 1/det(A) |

Methods to solve for Inverse | [A|I n] to find [I n|A -1] A -1 = 1/det(A) x adj(A) |

Properties of Matrix | (A+B)C = AC+BC C(A+B) = CA+CB (A+B)T = A T = B T (AB) T = B T x A T (KA) T = KA T (A T)T = A |

Methods to solve a Matrix | Gaussin-Elimiation (work to Row-Echeln form) Cramer's Rule Multiply [A] -1 x [b] |

Define Adjoin | adj(A) = [Cij]T Cij = (-1) i+j * Mij |

Define Cramer's Rule | X1 = det(A1)/det(A) X2 = det(A2)/det(A) X3 = det(A3)/det(A) |

Define Gaussine-Elimation | Using row elementry steps, reduce Matrix into Row-Echlen form. Leading 1, in each column follwed by 0 beneath. |

What is Augmented Matrix? | A# = [A|b vector] |

What is homogeneous? | Set of equations in which all equal 0 (0 on the right hand side). |

If an set of equations are homogenous, what does that tell us? | It has at LEAST one SOLUTION. (X1=0, X2=0, etc) |

In Gaussina-Elimation, what are the 3 possibilities? What else can be defferred? | 1-No solution. [000|8] 2-Infinite solution. [000|0] 3-Unique solution. (Not either of above) If Rank(A#)<# of variables, INFINITE solutions (free variables) 2 rows are same or multiples - INFINITE solutions |

Created by:
DrMolina