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Matrix Properites

Matrices, Determinents, Methods

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