Exam 1 Flashcards
Det(I) = ?
1
(A^T)^T = ?
A
(A + B)^T = ?
A^T + B^T
(AB)^T = ?
B^T x A^T
(AB)^-1 = ?
B^-1 x A^-1
(A^T)^-1 =
(A^-1)^T
AA^-1 = ?
I
What is the inverse of an elementary matrix?
The opposite operation.
Ex: - e.m. multiply r1 by 2;
inverse multiply r1 by 1/2
E3E2E1A = ? A = ?s
I
Inverse of E1E2E3
For any nxn matrix, the following (5) statements are equivalent:
(1) A is invertible
(2) Ax=b had a unique solution
(3) A is row-equivalent to I (can be converted using EROs
(4) Ax=0 only has the trivial solution
(5) A is a product of elementary matrices
What is prime factorization? Is it unique?
Factoring a matrix into a product of elementary matrices.
Factorization is not unique.
Determinants:
If B is obtained from A by switching two rows or columns then det(B) = ?
-det(A)
Determinants:
If C is obtained from A by multiplying a row or column by a scalar k, then det(C) = ?
kdet(A)
Determinants:
If D is obtained from A by adding a multiple of one row or column to another, then det(D) = ?
det(A)
Can determinants be found on any size matrix?
No, only nxn matrices.
If A and B are nxn, det(AB) = ?
det(A)det(B)
If A is not invertible, det(A) = ?
0
det(A^-1) = ?
1/[det(A)]
Det(A^T) = ?
det(A)
Vector spaces:
What is axiom (1)?
(1) If u and v belong to V, then u+v belongs to V
V is closed under addition
Vector spaces:
What is axiom (4)?
(4) The 0 vector belongs to V such that 0 + u = u
Vector spaces:
What is axiom (5)?
(5) For each u in V, there is a vector -u
s. t. u + (-u) = 0
Vector spaces:
What is axiom (6)?
(6) Closed under scalar multiplication
What subspaces of R^2 exist?
(1) Just 0
(2) Everything
(3) A straight line through the origin
Is the 0 matrix invertible?
No
What is a singular matrix?
A singular matrix is not invertible
it’s determinant is 0
There may be no solution or infinitely many solutions.
What is a non singular matrix?
An invertible matrix
Determinant = NONzero
One unique solution
What is an inconsistent system of equations?
One with no solution
Ex: x+y+z=1
x+y+z=12
Can all matrices be multiplied together?
No, inner numbers must match. Outer numbers tell what size the new matrix will be.
Ex: 2x2 and 2x3 will become 2x3
2x2 and 3x3 undefined
What does a row of all 0’s tell you?
Infinitely many solutions
What does a row of 0’s followed by a nonzero in an augmented matrix tell you?
No solution.
0x+0y cannot equal a nonzero number.
What is an augmented matrix?
A matrix containing the coefficients and the constant terms in a system of equations.
What is a coefficient matrix?
A matrix containing only the coefficients of a l.s.
What are the EROs?
- Interchange two rows
- Multiply a row by a NONZERO constant
- Add a multiple of one row to another
What is a homogenous system?
A l.s. in which all of the constant terms are 0
What is the trivial solution?
If all of the variables in a homogenous system are equal to zero
Is matrix multiplication commutative?
No!
AB is NOT EQUAL to BA
If A is an invertible nxn matrix and AB = AC, does B=C?
Yes, let B & C be inverses of A.
AB=I & AC=I
AB=I -> mult. both sides by C, reduce
B=C
