Matrix Algebra аnd Simultaneous Linear Equations. Lecture 10

LECTURE 9
MATRIX ALGEBRA AND SIMULTANEOUS LINEAR EQUATIONS
Temur Makhkamov
Indira Khadjieva
QM Module Leader
[e-mail]
Room IB 205

Lecture outline
The meaning and properties of matrices;
The arithmetic operations on matrices;
The applications of matrices to reality

Matrix
A Matrix is simply a rectangular array of numbers arranged in rows and columns.
The size of a matrix is indicated by the number of its rows and the number of its columns
The whole matrix is labeled by a capital letter
The individual numbers (elements) contained in the matrix are labeled by lower case letters with a suffix to identify their locations within the matrix.

Examples of matrices
Examples:
1) – 2x2 matrix
2) – 2x3 matrix
3) – ?

Addition (and Subtraction) of matrices
You can add (subtract) two matrices of the same size (equal number of rows and columns).
The sum (difference) of two equal-sized matrices results in the new matrix of the same size as the two matrices being added.
Example:

Exercise: Addition and Subtraction

Scalar multiplication
Multiply each element of the matrix by the number.
Example:

Exercise: Scalar multiplication

Matrix multiplication
Two matrices can be multiplied only if the number of columns of the 1st matrix equals to the number of the rows of the 2nd matrix.
Multiply rows of the 1st matrix by columns of the 2nd matrix
Example:

Matrix Transpose
The transpose of matrix can be obtained by interchanging the rows and columns
The first row of the matrix A is the first column of matrix A transposed

Zero & Identity matrix
Zero matrix is a matrix with all elements 0.
Identity matrix is a square matrix with elements of 1s on the main diagonal from top left to bottom right and 0s on other positions

Determinant of a matrix
A numerical value of matrix
Can be a negative number
Exists for a square matrix only
Determinant for 2x2 matrix is calculated as follows:

Inverse of a (2x2) matrix (1)
If then

Inverse of a (2x2) matrix (2)
Example: , then
Calculate:
Check:

Simultaneous linear equations

Equilibrium price in theory

Equilibrium price in practice

Equilibrium point in graph

Breakeven point in theory

Breakeven in practice

Breakeven point in graph

Three methods of solving SLE

Gauss’ method (elimination)

Matrix inverse method

Matrix inverse method

Cramer’s method