Chapter 2 Definitions Flashcards

1
Q

Definition of Equality of Matrices

A

equal when same size and aij = bij for 1

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2
Q

Definition of Matrix Addition

A

A + B = [aij + bij]

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3
Q

Definition of Scalar Multiplication

A

cA = [caij]

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4
Q

Definition of Matrix Multiplication

A

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

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5
Q

Linear Combinations of Column Vectors

A

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

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6
Q

Commutative property of addition

A

A + B = B + A

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7
Q

associative property of addition

A

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

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8
Q

associative property of multiplication

A

(cd)A = c(dA)

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9
Q

multiplicative identity

A

1A = A

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10
Q

distributive property

A

c(A + B) = cA + cB

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11
Q

distributive property

A

(c +d)A = cA + dA

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12
Q

property of zero matrix

A

A + Omn = A

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13
Q

property of zero matrix

A

A + (-A) = Omn

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14
Q

property of zero matrix

A

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

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15
Q

associative property of matrix multiplication

A

A(BC) = (AB)C

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16
Q

distributive property of matrix multiplication

A

A(B+C) = AB + AC

17
Q

distributive property of matrix multiplication

A

(A+B)C = AC + BC

18
Q

scalar properties of matrix multiplication

A

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

19
Q

property of identity matrix

A

AIn = A

20
Q

property of identity matrix

A

ImA = A

21
Q

properties of transposes

A

(A^T)^T=A

22
Q

properties of transposes

A

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

23
Q

properties of transposes

A

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

24
Q

properties of transposes

A

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

25
Q

Definition of the Inverse of a Matrix

A

AB = BA = In

26
Q

Properties of Inverse Matrices: 1

A

(A^-1)^-1=A

27
Q

Properties of Inverse Matrices: 2

A

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

28
Q

Properties of Inverse Matrices: 3

A

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

29
Q

Properties of Inverse Matrices: 4

A

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

30
Q

Inverse of a Product

A

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

31
Q

Definition of an Elementary Matrix

A

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

32
Q

Invertible Matrix Theroem: 1

A

A is invertible if A is an nxn matrix

33
Q

Invertible Matrix Theroem: 2

A

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

34
Q

Invertible Matrix Theroem: 3

A

Ax=0 has only the trivial solution

35
Q

Invertible Matrix Theroem: 4

A

A is row-equivalent to I

36
Q

Invertible Matrix Theroem: 5

A

A can be written as the product of elementary matrices