Matrix rules for Econometrics Flashcards
When are two matrices conformable in addition/subtraction?
When they both have the same number of rows (m) and columns (n)
What are the two main properties of Matrix addition/subtraction?
Commutative and Associative
What does Commutative in addition mean?
The order of the matrices is irrelevant:
A + B = B + A
What does Associative in addition mean?
The order in which matrices are added together does not matter:
A + (B + C) = (A + B) + C
When is the multiplication of two matrices conformable?
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
Steps of Matrix multiplication
Why is it possible that when multiplying two matrices:
AB ≠ BA
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
What is the reversal rule in matrix transposition?
Let C = AB. Matrix transposition flips the order of the product:
C’ = B’A’ which can be expressed as C = (AB)’
What is the result of a matrix being multiplied by the identity matrix?
For example: A x I
The original matrix - A
Hence, AI = IA = A
Steps for matrix inversion
What is AA^-1
The identity matrix I
What is the determinant of a matrix
When are two vectors orthogonal?
x’y = 0
Show that the inverse of an orthogonal matrix is its transpose:
A’ = A^-1
Importance of linear dependence and rank