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An introduction to matrices, their uses in mathematics, computing, and finance, and various operations such as scalar multiplication, addition, transpose, and matrix multiplication. It also covers the inverse of a matrix and its application in solving simultaneous equations.
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MSc Finance, MSc International Banking & Finance, MSc Investment & Finance, MSc International Accounting & Finance
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)