Exam 1

Question 1

Det(I) = ?

Answer 1

1

Question 2

(A^T)^T = ?

Answer 2

A

Question 3

(A + B)^T = ?

Answer 3

A^T + B^T

Question 4

(AB)^T = ?

Answer 4

B^T x A^T

Question 5

(AB)^-1 = ?

Answer 5

B^-1 x A^-1

Question 6

(A^T)^-1 =

Answer 6

(A^-1)^T

Question 7

AA^-1 = ?

Answer 7

I

Question 8

What is the inverse of an elementary matrix?

Answer 8

The opposite operation.

Ex: - e.m. multiply r1 by 2;
inverse multiply r1 by 1/2

Question 9

E3E2E1A = ? 
A = ?s

Answer 9

I

Inverse of E1E2E3

Question 10

For any nxn matrix, the following (5) statements are equivalent:

Answer 10

(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

Question 11

What is prime factorization? Is it unique?

Answer 11

Factoring a matrix into a product of elementary matrices.

Factorization is not unique.

Question 12

Determinants:

If B is obtained from A by switching two rows or columns then det(B) = ?

Answer 12

-det(A)

Question 13

Determinants:

If C is obtained from A by multiplying a row or column by a scalar k, then det(C) = ?

Answer 13

kdet(A)

Question 14

Determinants:

If D is obtained from A by adding a multiple of one row or column to another, then det(D) = ?

Answer 14

det(A)

Question 15

Can determinants be found on any size matrix?

Answer 15

No, only nxn matrices.

Question 16

If A and B are nxn, det(AB) = ?

Answer 16

det(A)det(B)

Question 17

If A is not invertible, det(A) = ?

Answer 17

0

Question 18

det(A^-1) = ?

Answer 18

1/[det(A)]

Question 19

Det(A^T) = ?

Answer 19

det(A)

Question 20

Vector spaces:

What is axiom (1)?

Answer 20

(1) If u and v belong to V, then u+v belongs to V

V is closed under addition

Question 21

Vector spaces:

What is axiom (4)?

Answer 21

(4) The 0 vector belongs to V such that 0 + u = u

Question 22

Vector spaces:

What is axiom (5)?

Answer 22

(5) For each u in V, there is a vector -u

s. t. u + (-u) = 0

Question 23

Vector spaces:

What is axiom (6)?

Answer 23

(6) Closed under scalar multiplication

Question 24

What subspaces of R^2 exist?

Answer 24

(1) Just 0
(2) Everything
(3) A straight line through the origin

Question 25

Is the 0 matrix invertible?

Answer 25

No

Question 26

What is a singular matrix?

Answer 26

A singular matrix is not invertible

it’s determinant is 0

There may be no solution or infinitely many solutions.

Question 27

What is a non singular matrix?

Answer 27

An invertible matrix

Determinant = NONzero

One unique solution

Question 28

What is an inconsistent system of equations?

Answer 28

One with no solution

Ex: x+y+z=1
x+y+z=12

Question 29

Can all matrices be multiplied together?

Answer 29

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

Question 30

What does a row of all 0’s tell you?

Answer 30

Infinitely many solutions

Question 31

What does a row of 0’s followed by a nonzero in an augmented matrix tell you?

Answer 31

No solution.

0x+0y cannot equal a nonzero number.

Question 32

What is an augmented matrix?

Answer 32

A matrix containing the coefficients and the constant terms in a system of equations.

Question 33

What is a coefficient matrix?

Answer 33

A matrix containing only the coefficients of a l.s.

Question 34

What are the EROs?

Answer 34

1. Interchange two rows
2. Multiply a row by a NONZERO constant
3. Add a multiple of one row to another

Question 35

What is a homogenous system?

Answer 35

A l.s. in which all of the constant terms are 0

Question 36

What is the trivial solution?

Answer 36

If all of the variables in a homogenous system are equal to zero

Question 37

Is matrix multiplication commutative?

Answer 37

No!

AB is NOT EQUAL to BA

Question 38

If A is an invertible nxn matrix and AB = AC, does B=C?

Answer 38

Yes, let B & C be inverses of A.

AB=I & AC=I

AB=I -> mult. both sides by C, reduce

B=C