Matrix rules for Econometrics

Question 1

When are two matrices conformable in addition/subtraction?

Answer 1

When they both have the same number of rows (m) and columns (n)

Question 2

What are the two main properties of Matrix addition/subtraction?

Answer 2

Commutative and Associative

Question 3

What does Commutative in addition mean?

Answer 3

The order of the matrices is irrelevant:
A + B = B + A

Question 4

What does Associative in addition mean?

Answer 4

The order in which matrices are added together does not matter:
A + (B + C) = (A + B) + C

Question 5

When is the multiplication of two matrices conformable?

Answer 5

When the number of columns for the first matrix equals the number of rows of the second matrix:
(A: m x n)(B: n x p) = (C: m x p)

Where C = AB is the matrix product

Question 6

Steps of Matrix multiplication

Question 7

Why is it possible that when multiplying two matrices:
AB ≠ BA

Answer 7

Matrix multiplication is not commutative which means that that the order in which two matrices are multiplied matters. The case where AB = BA is only possible when matrices A and B have equal numbers of rows and columns

Question 8

What is the reversal rule in matrix transposition?

Answer 8

Let C = AB. Matrix transposition flips the order of the product:
C’ = B’A’ which can be expressed as C = (AB)’

Question 9

What is the result of a matrix being multiplied by the identity matrix?
For example: A x I

Answer 9

The original matrix - A
Hence, AI = IA = A

Question 10

Steps for matrix inversion

Question 11

What is AA^-1

Answer 11

The identity matrix I

Question 12

What is the determinant of a matrix

Question 13

When are two vectors orthogonal?

Answer 13

x’y = 0

Question 14

Show that the inverse of an orthogonal matrix is its transpose:
A’ = A^-1

Question 15

Importance of linear dependence and rank