Prepatory Materials

Question 1

Diagonal matrix

Answer 1

A square matrix where the leading diagonal elements are non-zero but all the off diagonal elements are zero

Question 2

Null matrix

Answer 2

All values are zero

Question 3

When does matrix multiplication work?

Answer 3

When the number of columns in A equals the number of rows in B

Question 4

What does taking a transpose do?

Answer 4

Converts the rows into columns

Question 5

What is (A’)’ equal to?

Answer 5

A

Question 6

What is (A+B)’ equal to?

Answer 6

A’ +B’

Question 7

What is (AB)’ equal to?

Answer 7

B’A’

Question 8

Symmetric matrix

Answer 8

One that is unchanged when transposed

Question 9

Trace

Answer 9

The sum of the diagonal elements in a matrix

Question 10

How do you find the determinant of higher order matrices?

Answer 10

Define the minor Mij as the determinant of the submatrix obtained by removing the ith row and the jth column from A, and define the corresponding cofactor. The determinant is given by the sum of all elements and cofactors

Question 11

What is |A’| equal to?

Answer 11

|A|

Question 12

What is |AB| equal to?

Answer 12

|A| x |B|

Question 13

What is |In| equal to?

Answer 13

In

Question 14

What is |kA|equal to?

Answer 14

|kA| = k^n|A|

Question 15

Which matrices have an inverse?

Answer 15

All non singular matrices

Question 16

What are the four equations a generalised inverse (Moore- Penrose inverse) must satisfy?

Answer 16

• AA^(+)A=A
• A^(+)AA^(+)=A^(+)
• (AA^(+))’ = AA^(+)
• (A^(+)A)’ = A^(+)A

Question 17

What is rank?

Answer 17

The concept of linear dependence among vectors. The rank of a matrix is equal to the maximum number of linearly independent rows or columns

Question 18

What happens to a square matrix’s determinant if it has less than full rank?

Answer 18

The determinant will be zero and therefore the matrix will be singular

Question 19

Orthogonal

Answer 19

Two vectors are orthogonal if their inner product is zero

Question 20

When does the multiple solution case of interest arise with eigenvalues?

Answer 20

Where A- lambda x identity matrix is singular

Question 21

What is the product of the eigenvalues equal to in a symmetric matrix?

Answer 21

The determinant

Question 22

What is the sum of the eigenvalues equal to in a symmetric matrix?

Answer 22

The trace of the matrix

Question 23

Idempotent matrix

Answer 23

A square matrix which reproduces itself under multiplication ie A=AA

Question 24

What do the eigenvalues of a symmetric idempotent matrix equal?

Answer 24

0 or 1

Question 25

What does the determinant of a symmetric idempotent matrix equal?

Answer 25

0 or 1

Question 26

What does the trace of a symmetric idempotent matrix equal?

Answer 26

The number of non zero eigenvalues

Question 27

Which symmetric idempotent matrices are singular?

Answer 27

All of them apart from the identity matrix

Question 28

When is A positive definite?

Answer 28

In a quadratic where if x’Ax >0 for all values of x apart from x =0

Question 29

When is A negative definite?

Answer 29

In a quadratic where if x’Ax <0 for all values of x apart from x =0

Question 30

When is A positive semi definite?

Answer 30

In a quadratic where if x’Ax >=0 for all values of x apart from x =0

Question 31

When is A negative semi definite?

Answer 31

In a quadratic where if x’Ax <=0 for all values of x apart from x =0

Question 32

What is E(ax+b) equal to?

Answer 32

aE(x) +b

Question 33

What is V(ax+b) equal to?

Answer 33

a^2 V(x)

Question 34

Equation for covariance

Answer 34

Cov(xi,xj) = E((xi-E(xi))(xj-E(xj)))

Question 35

When is the joint density of x given by the product of the marginal densities?

Answer 35

When the variables in x are independent

Question 36

What is the square of the t stat the same as?

Answer 36

An F stat with degrees of freedom 1 and v

Question 37

Chebychevs inequality

Answer 37

Pr(|x-E(x)| >=ę) =< V(x)/ę^2 Where x is any random variable