Matrices and Systems of Equations Flashcards
Define a field
A structure where we can
1) Add, subtract and multiply any two elements and divide any element by any non-zero element and in doing so stay in the field
2) The natural rules of arithmetic hold
What is the transpose of a matrix?
A n x m matrix whose (i,j)th entry is aji
(AB)^T
B^T*A^T
What must be true for a matrix to be in echelon form?
1) Any zero rows appear at the bottom of A
2) Each leading entry (first non-zero entry in a row) is in a column to the right of the leading entry in the row above it
What must also be true on top of echelon conditions for a matrix to be in reduced echelon form?
3) In each non-zero row, the leading entry is 1
4) In each column with a leading entry, all other entries are zero
An n x n matrix is symmetric if….
A^T = A
An n x n matrix is skew symmetric if…..
A^T = -A
An n x n matrix is idempotent if….
A^2 = A
A ~ B if…
There is a sequence of e.r.o’s that transform A into B
number of unknowns - number of equations
number of parameters needed
A square matrix P is orthogonal if ….
P-1 = P^T
What statements are equivalent with A being invertible?
1) For each n x 1 column vector b, then the system Ax = b has a unique solution
2) A is row equivalent to In
3) det(A) != 0
What does it mean for a system of linear equations to be consistent?
If it has one or more solutions
What does it mean for a system of equations to be inconsistent?
If it does not have one or more solutions
What is the relationship between det(B) and det(A) if you obtain B from swapping two rows in A?
det(B) = - det(A)
