Matrices and Systems of Equations Flashcards Preview

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Flashcards in Matrices and Systems of Equations Deck (19)
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1
Q

Define a field

A

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

2
Q

What is the transpose of a matrix?

A

A n x m matrix whose (i,j)th entry is aji

3
Q

(AB)^T

A

B^T*A^T

4
Q

What must be true for a matrix to be in echelon form?

A

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

5
Q

What must also be true on top of echelon conditions for a matrix to be in reduced echelon form?

A

3) In each non-zero row, the leading entry is 1

4) In each column with a leading entry, all other entries are zero

6
Q

An n x n matrix is symmetric if….

A

A^T = A

7
Q

An n x n matrix is skew symmetric if…..

A

A^T = -A

8
Q

An n x n matrix is idempotent if….

A

A^2 = A

9
Q

A ~ B if…

A

There is a sequence of e.r.o’s that transform A into B

10
Q

number of unknowns - number of equations

A

number of parameters needed

11
Q

A square matrix P is orthogonal if ….

A

P-1 = P^T

12
Q

What statements are equivalent with A being invertible?

A

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

13
Q

What does it mean for a system of linear equations to be consistent?

A

If it has one or more solutions

14
Q

What does it mean for a system of equations to be inconsistent?

A

If it does not have one or more solutions

15
Q

What is the relationship between det(B) and det(A) if you obtain B from swapping two rows in A?

A

det(B) = - det(A)

16
Q

What is the relationship between det(B) and det(A) if you obtain B from scaling a row with a?

A

det(B) = a*det(A)

17
Q

What is the relationship between det(B) and det(A) if you obtain B by subtracting from one row a non-zero scalar multiple of another?

A

det(B) = det(A)

18
Q

det(AB) =

A

det(A)*det(B)

19
Q

If A is invertible, then A-1 is

A

(1/det(A)) * adj(A)