Matrices Flashcards
Singular matrix
the determinant is zero
Symmetric matrix
when the matrix equals its own transpose
Trace of a matrix
the sum of the elements of the main diagonal
(square matrices only)
Inconsistent system of linear equations
it has no solutions
Consistent system of linear equations
it has a solution
Homogenous system
a system of linear equations having matrix form AX=0 (0 represents a zero column matrix)
Inhomogenous system
a system of linear equations having matrix form AX=b (b is non-zero column vector)
When are a system of equations linearly dependent?
when one row is a multiple of another or the sum of multiples of other rows
Commutative property
order changes but answer doesn’t
Associative property
(grouping) changes but answer doesn’t
Commutative example
A+B=B+A
Associative example
(A+B)C=A(B+C)
What are the steps for finding the inverse of a 3x3 matrix?
- find detA
- find the matrix of minors (M)
- find the matrix of cofactors (C)
- transpose the matrix of cofactors (C^t)
- use 1/ detA C^t to find A^-1
What is the matrix of minors?
you replace every element in a matrix by its minor
What is the matrix of cofactors?
including the sign changes (+-+-)
How do you transpose the matrix of cofactors?
Interchange the columns and rows so first row becomes first column, etc
How do you know if a system of linear equations is consistent or not?
if it is consistent, there will be at least one set of values that satisfies all of the equations simultaneously. If not, they are inconsistent.
How do you know if equations will be consistent or not when det m = 0
- use two equations and cancel out a term to give 2 different, 2 term equations.
- if you can simplify or multiply them by a number to make them the same, they are consistent.
- if you can make the coefficients the same but not the answers, they are inconsistent
Generally for a system of linear equations, what does D≠0 mean for the solution?
there is a single, unique solution
