Matrices in Finance: Size, Shape, and Operations, Lecture notes of Finance

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MSc Finance, MSc International Banking & Finance, MSc Investment & Finance, MSc International Accounting & Finance

Computing for Finance

and Financial Analysis

(Excel)

Lecture 6 - Week 6

Matrices A matrix is a table or array of numbers or variables Many uses in mathematics, computing and finance Examples include solving linear simultaneous equations and calculating efficient frontiers

Example (a) 3 rows x 2 columns

Example (b) 3 rows x 3 columns 3x3 square matrix

Example (d) 3 rows x 1 column 3x1 matrix a column vector

Example (e) 4 rows x 4 columns 4x4 square matrix 4x4 Identity matrix (I)

Scalar Multiplication A number * a matrix Result is the same shape as the original matrix Multiply each element in the matrix by the scalar

Scalar Multiplication In Excel: Multiply each cell in the matrix e.g. =B22 etc. Use an array formula: {=B2:C42}

Transpose Changes rows into columns and columns into rows Denoted by T or ‘ Use the =TRANSPOSE array formula This is not the same as Transpose from Paste Special

Matrix Multiplication Size and shape matters Example: A = 3x2, C = 2x 3x2 2x If these are equal, multiplication is possible Result is shape of outer two numbers

Inverse of a Matrix Inverse of a matrix A is defined as: AA*

- = I = A - *A Always a square matrix =MINVERSE(range) Can be used to solve simultaneous equations

Simultaneous Equations Translate the equations into matrices The result vector X is then the inverse of A multiplied by the vector Y X = A

- *Y =MMULT(MINVERSE(A-range),Y-range)