Math150 (Linear Algebra, 线性代数) Final Exam Formula sheet

M ATH 150 F INAL EXAM F ORMULA SHEET

(by: Dr. Yu)

Rules of Matrix Arithmetic:

A +

B =B +A

(Commutative Law of Addition)

A +(

B +C)=(A +B)+

C (Associated Law of Addition)

A(BC)=(AB)C

(Associative Law of Multiplication)

()±=±A B C AB AC (Distributive Law)

a a a (B +C)=B +C ()()a

b a b =C C ()a b a b ±=±C C C

()()()a a a ==BC B C B C

(Note: Usually ≠AB BA )

A +O =O +A A -A =O

O -A =-A AO =O

OA =O

AI =IA =A (very important!)

(O is a zero matrix, and I is an identity matrix)

Inverse Rules:

-1-1A A =AA =I (very important!) 111()---=AB B A 11()--=A A

111

()k k --=A A

1

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

Power Rules:

p q p q +=A A A ()p q pq =A A for example: 2=A AA 422=A A A 54=A A A

102552()()==A A A

Some Remarkable properties of Matrices:

≠AB BA (usually) 2=A O but ≠A O

AB =O but ≠A O and ≠B O 2=A I but ≠A I and ≠-A I (p p p p p ≠≠AB)A B B A (usually) (T T T T T =≠AB)B A A B 11111()-----=≠AB B A A B

Transpose Rules:

()()()()T T T T T

T

T

T T T αα===+=A A

A A A +

B A B

AB B A

If T =A A then A is a symmetric matrix.

11()()T T --=A A

Graph Theory:

1. Strong Connection:

231n -+A +A +A +...A =M , and all 0ij m >.

2. Clique: 3(3)ij s ??=??S , and (3)

0ii s > .

● Leontief closed model:

() = -A I p 0

● Leontief open model:

(I -C) x = d

C is “productive” if 1-(I -C) exists, and all entries 0≥.

Determinants and Solution of Linear Systems

● For inhomogeneous system: Ax =b ,

where ≠b 0

a) det()0A ≠: There is a unique solution.

b) det()0A =: There is no solution or

infinite number of solutions.

● For homogeneous system: Ax =0

a) det()0A ≠: There is no nontrivial

solution; the only solution is the trivial solution x =0 12(...0)n x x x ====.

b) det()0A =: There exist nontrivial

solutions (in fact, infinite number of solutions).

Properties of Determinants

● 22?determinants:

a b

ad bc c d =-

● 33? determinants:

● det()det()T =A A

● Interchange two rows (or columns) →

the determinant change sign.

● If two rows (or columns) are equal →

the determinant = 0

● If a row (or column) contains all zeros

→ determinant = 0

● Multiply a real number c to any row or

(column) of det()A → the new determinant = c det()A .

● The elementary row operation of the

type “122r k r r ?+→” does not change the value of the determinant, where k is a non-zero constant. It is also true for the same type of column operation.

● For any triangular matrix

112233det()...nn A a a a a =

The Cofactor Expansion

1

det()n

ij ij j A a A ==

∑,

where (1)i j ij ij A M +=-, ij A is so called the cofactor of

ij a , and ij M is the (,)i j th “minor” of matrix A .

Eigenvalues , eigenvectors :

For the linear system:

λ=Ax x ?()λ-=A I x 0 or ()λ-=I A x 0

(homogeneous systems.) a) I f det()0λ-=A I

there exist nontrivial solutions.

b)det()0λ-=A I is called characteristic equations .

c) The numbers i λ which satisfies the characteristic equations are called the eigenvalues .

d) The solutions of the linear system correspond to eigenvalues are called the eigenvectors .

Properties of eigenvalues

If 12,,...,n λλλare the eigenvalues of the matrix A , then 12det()n λλλ=???A . If 12,,...,n λλλbe the eigenvalues of the matrix A , then 12,,...,k k k n λλλare the eigenvalues of the matrix k A . If 12,,...,n λλλare the eigenvalues of the matrix A , then 12,,...,n c c c λλλ+++are the eigenvalues of the matrix c +A I .

The“trace”of the matrix A , i.e., the sum of the elements on the main diagonal, is equal to the sum of all eigenvalues of the matrix A .

Linear Independence

The vectors 12,,...,k v v v are said to be linearly dependent if there exist constants 12,,...,k c c c , not all zero, such that 1122...k k c c c +++=v v v 0. Otherwise, 12,,...,k v v v are said to be linearly independent that means the only solution must be the trivial solution: 12...0k c c c ====.