Maths for Economics 8: Vectors And Matrices In Economics And Finance Topic 2 - Matrix Fundamentals

Question 1

What’s the rule about Matrix times Vector?

Answer 1

The number of columns of the matrix must equal the number of rows of the vector,
or the multiplication cannot be performed.

Question 2

What’s the rule about Matrix Multiplication?

Answer 2

Number of columns of former matrix = Number of rows of latter matrix
or the multiplication cannot be performed. If E is m x n and F is p x q, then the product EF exists only if n=p,
in which case is EF is m x q.

Question 3

Describe the terms ‘commutative’ and ‘associative’ and apply them to matrix addition and multiplication

Answer 3

Matrix addition is commutative because A+B = B+A (always).
It is associative because A+(B+C) = (A+B)+C (always).
Matrix multiplication is non-commutative because AB does not equal BA, except
in special cases.
However matrix multiplication is associative because A(BC) = (AB)C (always)

Question 4

Describe & explain ‘Transposes’

Answer 4

If A = [ 1 2 3 (column 1) 6 5 4 (column 2), then the transpose of A is A^T = [ 1 2 3 (row 1) 6 5 4 (row 2).
Element (i, j) of A is the same as element (j, i) of A^T.
There are two important rules for dealing with transposes:
(1) Transpose of a Product. (AB)^T = B^T A^T, (and
(ABC)^T = C^T B^T A^T etc.) The order in which the matrices occur is
reversed.
(2) Transpose of a Transpose. (A^T)^T = A. Transposing is a ‘toggle’
operation.