Chapter 2 Definitions

Question 1

Definition of Equality of Matrices

Answer 1

equal when same size and aij = bij for 1

Question 2

Definition of Matrix Addition

Answer 2

A + B = [aij + bij]

Question 3

Definition of Scalar Multiplication

Answer 3

cA = [caij]

Question 4

Definition of Matrix Multiplication

Answer 4

AB = [cij] where cij = sum(k=1, n, aikbkj)

Question 5

Linear Combinations of Column Vectors

Answer 5

x11a1 + x21a2 + … + xnan = b, where A is matrix mxn, x is nx1, b nx1

Question 6

Commutative property of addition

Answer 6

A + B = B + A

Question 7

associative property of addition

Answer 7

A + (B + C) = (A + B) + C

Question 8

associative property of multiplication

Answer 8

(cd)A = c(dA)

Question 9

multiplicative identity

Answer 9

1A = A

Question 10

distributive property

Answer 10

c(A + B) = cA + cB

Question 11

distributive property

Answer 11

(c +d)A = cA + dA

Question 12

property of zero matrix

Answer 12

A + Omn = A

Question 13

property of zero matrix

Answer 13

A + (-A) = Omn

Question 14

property of zero matrix

Answer 14

if cA = Omn then c=0 or A = Omn

Question 15

associative property of matrix multiplication

Answer 15

A(BC) = (AB)C

Question 16

distributive property of matrix multiplication

Answer 16

A(B+C) = AB + AC

Question 17

distributive property of matrix multiplication

Answer 17

(A+B)C = AC + BC

Question 18

scalar properties of matrix multiplication

Answer 18

c(AB)=(cA)B=A(cB)

Question 19

property of identity matrix

Answer 19

AIn = A

Question 20

property of identity matrix

Answer 20

ImA = A

Question 21

properties of transposes

Answer 21

(A^T)^T=A

Question 22

properties of transposes

Answer 22

(A+B)^T=A^T+B^T

Question 23

properties of transposes

Answer 23

(cA)^T=c(A^T)

Question 24

properties of transposes

Answer 24

(AB)^T=B^T*A^T

Question 25

Definition of the Inverse of a Matrix

Answer 25

AB = BA = In

Question 26

Properties of Inverse Matrices: 1

Answer 26

(A^-1)^-1=A

Question 27

Properties of Inverse Matrices: 2

Answer 27

(A^k)^-1 = (A^-1)^k

Question 28

Properties of Inverse Matrices: 3

Answer 28

(cA)^-1=c^-1A^-1

Question 29

Properties of Inverse Matrices: 4

Answer 29

(A^T)^-1=(A^-1)^T

Question 30

Inverse of a Product

Answer 30

(AB)^-1=B^-1*A^-1

Question 31

Definition of an Elementary Matrix

Answer 31

nxn where it can be obtained from the identity matrix In by a single elementary row operation

Question 32

Invertible Matrix Theroem: 1

Answer 32

A is invertible if A is an nxn matrix

Question 33

Invertible Matrix Theroem: 2

Answer 33

Ax=b has a unique solution for every nx1 column matrix b

Question 34

Invertible Matrix Theroem: 3

Answer 34

Ax=0 has only the trivial solution

Question 35

Invertible Matrix Theroem: 4

Answer 35

A is row-equivalent to I

Question 36

Invertible Matrix Theroem: 5

Answer 36

A can be written as the product of elementary matrices