Matrices

Question 1

When can guassian elimination be done

Answer 1

when the number of unknowns equal the number of equations (square matrix)

Question 2

If ax=b, what does:
1. a≠0 (b doesn’t matter here)
2. a=0, b≠0
3.a=0, b=0
equal

Answer 2

1. unique solution
2. no solution
3. infinite solutions

Question 3

When can matrix addition occur

Answer 3

when both matrices have the same order

Question 4

when can you multiply matrices A and B

Answer 4

when the number of columns of A are equal tot he number of rows of B

Question 5

What is a zero matrix

Answer 5

all elements of matrix are 0
A + 0 =A

Question 6

Properties: (λA)B

Answer 6

=λ(AB)
=A(λB)

Question 7

Properties: A(BC)

Answer 7

=(AB)C order of matrices must stay same

Question 8

Properties: (A+B)C

Answer 8

=AC + BC

Question 9

Properties: C(A+B)

Answer 9

=CA + CB

Question 10

Properties: true or false AB=BA

Answer 10

False: AB≠BA (usually) and AB-BA≠0

Question 11

Properties: true or false AB=0 means A=0 or B=0

Answer 11

False: AB=- does not mean A=0 or B=0

Question 12

Properties: A+B-C

Answer 12

=B-C+A

Question 13

Properties: (A+B)(A-B)

Answer 13

≠AA-BB

Question 14

What is the identity matrix

Answer 14

Square
leading diagonoal is 1, everything else is 0
Acts like the number 1
AI=A
Ix=x (where x is a column vector with same number of rows as I)

Question 15

Transpose D=EF

Answer 15

Dᵀ=FᵀEᵀ

Question 16

What is a transposed matrix

Answer 16

columns become equivalent rows and vice versa

Question 17

What matrices can have an inverse

Answer 17

square
when the determinant isn’t 0

Question 18

What does AA⁻¹ equal

Answer 18

=A⁻¹A=I

Question 19

How to show A⁻¹ is inverse A

Answer 19

Ax=b
AA⁻¹x=A⁻¹b
Ix=A⁻¹b
x=A⁻¹b

Question 20

Inverse D=EF

Answer 20

D⁻¹=F⁻¹E⁻¹

Question 21

show proof that inverse EF is F⁻¹E⁻¹

Answer 21

F⁻¹E⁻¹EF
=F⁻¹IF
=F⁻¹F
=I

proof as AA⁻¹=I

Question 22

What does invertible mean

Answer 22

A square matrix that has an inverse

Question 23

What matrices dont have inverses

Answer 23

Non-square
Zero-matrices (det=0 for 0-matrices)

Question 24

What is the inverse of an identity matrix

Answer 24

The identity matrix (itself)

Question 25

What is an orthogonal matrix

Answer 25

A rotation in 3D A matrix whose inverse equals its transpose

Question 26

What is a symmetric matrix

Answer 26

A=Aᵀ values below and above the leading diagonal are mirror opposites

Question 27

What is an anti-symmetric matrix

Answer 27

A=-Aᵀ Values below the leading diagonal are negative mirror opposites of the values above Leading diagonal are all 0

Question 28

What are the properties of orthogonal matrices

Answer 28

A⁻¹=Aᵀ If orthogonal: AAᵀ=I

Question 29

How do you find the determinant of a 2x2 mtrix

Answer 29

D=|a₁₁ a₁₂| |a₂₁ a₂₂| Times the lead diagonal, minus the multiplication of reverse diagonal D=a₁₁a₂₂ - a₁₂a₂₁

Question 30

How do you work out the inverse of a 2x2 system

Answer 30

A⁻¹=1/D |a₂₂ - a₁₂| |-a₂₁ a₁₁|

Question 31

What does it mean if D=0

Answer 31

Matrix is singular/has no inversea

Question 32

How do you wokr out the determinant of a 3x3 matrix

Answer 32

D=a₁₁|a₂₂ a₂₃| - a₁₂ |a₂₁ a₂₃|+a₁₃|a₂₁ a₂₂| |a₃₂ a₃₃| |a₃₁ a₃₃| |a₃₁ a₃₂|

Question 33

Properties: det(Aᵀ)=

Answer 33

=detA

Question 34

properties: |λa λb| |c d |

Answer 34

=λ|a b| |c d| If a factor λ appears in each element of a row or column, it can be taken out as a factor

Question 35

properties: |0 0| |c d|

Answer 35

=0 If all elements of a row or column are zero, determinant is zero

Question 36

properties: Det(λA) where A is mxn matrix

Answer 36

=λⁿdet(A)

Question 37

Properties: Det(-A) where A is mxn matrix

Answer 37

=(-1)ⁿdet(A)

Question 38

Properties:|a b| |c d|

Answer 38

= - |c d| |a b|

Question 39

Properties: |λc λd| |c d |

Answer 39

=λ|c d| = 0 |c d| If any row is a mulitple of another row, det=0

Question 40

Properties: |a+ λc b+ λd| | c d |

Answer 40

=|a b| |c d| Adding a multiple of one row to another, determinant doesn't change

Question 41

Properties: If A and B are square matrixes of the same order, det(AB)=

Answer 41

=det(A)det(B) If A=I AB=IB=B det(AB)=Det(B)=det(I)det(B)=det(A)det(B)

Question 42

What is the gauss-jordan method

Answer 42

Used to find inverse of 3x3 matrix (A:I) matrix a augmented with i, solve to have (I:B) where B is A⁻¹

Question 43

What is the eignevalue equation

Answer 43

Ax=λx Ax-λx=0 Ax-λIx=0 (A-λI)x=0 A is square x is column, non-zero, **eigenvector** λ is scalar multiplier, **eigenvalue**

Question 44

What does it mean if det(A-λI)≠0

Answer 44

invertible/non-singular as if det(a)=0, sngular x=(A-λI)⁻¹0=0 (eigenvector is 0) There are no non-trivial solutions for λ

Question 45

What does it mean if det(A-λI)=0

Answer 45

singular/non-invertible x≠0 NON-ZERO SOLUTION

Question 46

How many eigenvectors correspond to any given eigenvalue and how

Answer 46

Infinite Ax=λx A(cx)=cAx=c(λx) =λ(cx) where c is non-zero scalar so infinte number of solutions to linear system

Question 47

How do you solve for the eigenvalue

Answer 47

Find det(A-λI) and solve CHARACTERSITIC POLYNOMIAL

Question 48

How do yo solve for eigenvectors

Answer 48

Augment matrix A with 0's and make base row 0's with ERO's to create 2 simultaneous equations to solve and put in terms of t

Question 49

What do the product of eigenvalues produce

Answer 49

λ₁λ₂λ₃=detA

Question 50

For orthogonal matrix, has eigenvalue λ, what does xxᵀ=

Answer 50

=xx=|x|²

Question 51

When inverting matrices, when would you get no solution

Answer 51

Value created by the last row of an augmented matrix (to 0 on RHS) from ERO's creates a contradiction with other rows

Question 52

When inverting matrices, when would you get a unique solution

Answer 52

Values created by the augmented matrix (to 0 on RHS) from ERO's creates a unique coeffieicient for each varibale whe certain varible cannot occur e.g.

Question 53

When inverting matrices, when would you get infinite solutions

Answer 53

Value created by the last row of an augmented matrix (to 0 on RHS) from ERO's become a dependent system/some equations have duplicates of are linear combos of eachother so don't add any further contraints you have fewer independent equations than unknowns you need last row to have a zero coefficient to have infinite solutions e.g.4+k=0 when k=-4

Question 54

Can λ be 0

Answer 54

Yes

Question 55

Can an eigenvector be 0

Answer 55

no

Question 56

Prove that MMᵀ is symmetric

Answer 56

(MMᵀ)ᵀ=(Mᵀ)ᵀMᵀ=MMᵀ