Exam 1

Question 1

tr(A)

Answer 1

sum of main diagonal of square matrix

Question 2

row equivalent

Answer 2

you can get from A to B using only elementary row operations

Question 3

(A^-1)^-1 =

Answer 3

A

Question 4

(A^n)^-1 =

Answer 4

A^-n = (A^-1)^n

Question 5

(αA)^-1 =

Answer 5

(1/α)A^-1

Question 6

A^r A^s =

Answer 6

A^(r+s)

Question 7

(A^r)^s =

Answer 7

A^(rs)

Question 8

(A^T)^T =

Answer 8

A

Question 9

(A+B)^T =

Answer 9

A^T+B^T

Question 10

(αA)^T =

Answer 10

αA^T

Question 11

(AB)^T =

Answer 11

B^T A^T

Question 12

If you get a zero row while inverting a matrix,

Answer 12

it is not invertible

Question 13

A is symmetric if

Answer 13

A^T=A

Question 14

M_i,j(A) =

Answer 14

det of matrix gotten from A by removing row i and column j.

Question 15

C_i,j(A) =

Answer 15

(-1)^(i+j) M_i,j(A)

Question 16

det(αA) =

Answer 16

α^n det(A)

Question 17

det(A+B) ≠

Answer 17

detA+detB

Question 18

detAB =

Answer 18

detAdetB

Question 19

Biggerer Theorem

Answer 19

(a) A is invertible
(b) Ax=0 has only trivial solution
(c) A is row equivalent to I
(d) A is a product of elementary matrices
(e) Ax=b is consistent for any b
(f) Ax=b has only one solution for any b
(g) detA≠0

Question 20

detA^-1 =

Answer 20

1/detA

Question 21

adj(A) =

Answer 21

(CofA)^T

Question 22

Aadj(A) =

Answer 22

(detA)I

Question 23

Cramer’s Rule

Answer 23

For Ax=b and A is square, if detA≠0 then x_i=det(A_i)/det(A) where A_i=A with column i replaced by b.

Question 24

A vector is defined by

Answer 24

its end point minus its start point

Question 25

u+v =

Answer 25

v+u

Question 26

u+(v+w) =

Answer 26

(u+v)+w

Question 27

u+0=

Answer 27

u

Question 28

u-u =

Answer 28

0

Question 29

α(u+v)=

Answer 29

αu+αv

Question 30

(α+β)u=

Answer 30

αu+βu

Question 31

α(βu)=

Answer 31

(αβ)u

Question 32

u·v =

Answer 32

v·u

Question 33

u·(v+w) =

Answer 33

u·v + u·w

Question 34

α(u·v) =

Answer 34

(αu)·v = u·(αv)

Question 35

Norm (length) of v =

Answer 35

||v|| = √(v·v)

Question 36

||αv|| =

Answer 36

|α| ||v||

Question 37

normalization of v =

Answer 37

(1/||v||)v

Question 38

distance d(u,v)=

Answer 38

||u-v|| = ||v-u||

Question 39

cosθ =

Answer 39

u·v if u and v are unit vectors

Question 40

Cauchy-Schwartz Inequality

Answer 40

|u·v| ≤ ||u|| ||v||

Question 41

u and v are orthogonal if

Answer 41

u·v = 0

Question 42

(proj_u)v =

Answer 42

(u·v)u if u is a unit vector

Question 43

(proj_u^⊥)v =

Answer 43

v - (proj_u)v

Question 44

Line through point P in the direction of v

Answer 44

r(t) = P + tv

Question 45

Plane through point P in the direction of u and v

Answer 45

x(s,t) = P + su + tv