1- Matrix algebra

Question 1

How are matrices denoted?

Answer 1

Bold capital letters

Question 2

How are vectors denoted?

Answer 2

Bold lowercase letters, we assume they are column vectors

Question 3

When are 2 matrices conformable in addition?

Answer 3

When they have the same number of rows and columns

Question 4

What are the 2 main properties of matrix addition?

Answer 4

Commutative & Associative

Question 5

What does Commutative in addition mean?

Answer 5

It doesn’t matter which matrix is first

Question 6

What does Associative in addition mean?

Answer 6

With several matrices it doesn’t matter which terms are added first

Question 7

What does transposing do?

Answer 7

Transposing swaps rows and columns i.e. first row becomes first column etc

Question 8

What is the reversal rule for transposing? C=AB

Answer 8

C’ = B’A’ = (AB)’

Question 9

What happens when any matrix is multiplied by the identity matrix I?

Answer 9

It stays the same AI = IA = A

Question 10

What is the result of an inverse square matrix multiplied by itself?

Answer 10

Identity matrix

Question 11

What are the 3 steps to invert a square matrix?

Answer 11

1. Swap principal diagonal elements
2. Make the other diagonal negative
3. Multiply by the determinant

Question 12

What is the determinant of a square matrix?

Answer 12

1 over product of principal diagonal minus product of secondary diagonal

Question 13

What is the inverse of an inverse matrix?

Answer 13

The original

Question 14

What is the transpose of an inverse?

Answer 14

The inverse of the transpose, they are interchangeable

Question 15

What is the reversal rule for inverting matrices?

Answer 15

(AB)⁻¹ = B⁻¹A⁻¹

Question 16

When are two vectors orthogonal?

Answer 16

x’y = 0

Question 17

How could you expand the variance expression var(Ax)?

Answer 17

Avar(x)A’

Question 18

How could you expand the variance expression var(X-U)?

Answer 18

var(X) + var(U) - 2cov(X,U)

Question 19

When are matrices conformable in multiplication?

Answer 19

When the number of columns of the first one equals the number of rows of the second (n x r)x(r x m)

Question 20

What are the dimensions of the product of two matrices?

Answer 20

Rows of the first by columns of the second e.g.
(n x r)x(r x m) = (n x m)

Question 21

How do you multiply matrices?

Answer 21

Apply columns of the second to rows of the first i.e. first element will be the sum of products of first row and first column

Question 22

How do you show a matrix is symmetric?

Answer 22

Transpose equals itself A’=A

Question 23

How do you show a matrix is idempotent?

Answer 23

Original equals the square AA=A

Question 24

What is the trace of a square matrix?

Answer 24

Sum of the elements on the principle diagonal

Question 25

How can you show a matrix is idempotent using its trace?

Answer 25

trace(A) = rank(A)

Question 26

What are the 4 trace operating properties for square matrices?

Answer 26

1.tr(A+B) = tr(A) + tr(B)
2.tr(A) = tr(A’)
3.tr(kA) = ktr(A)
4.tr(AB) = tr(BA)

Question 27

How do you find the rank of a matrix?

Answer 27

1.Check if any row or column can be linearly expressed in terms of another
2.Take out the dependent vector
3.If the remaining vectors are independent, the number of them is the rank

Question 28

What is a full rank matrix?

Answer 28

All vectors are linearly independent

Question 29

What are the dimensions of X?

Answer 29

n x k

Question 30

What is the formula to prove linear dependence?

Answer 30

Σγᵢxᵢ = 0

Question 31

What is the transpose of a diagonal matrix?

Answer 31

Itself Ω’ = Ω
Diagonal matrices are symmetrical