Matrix Algebra аnd Simultaneous Linear Equations. Lecture 10

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LECTURE 9
MATRIX ALGEBRA AND SIMULTANEOUS LINEAR EQUATIONS
Temur Makhkamov
Indira Khadjieva
QM Module Leader

LECTURE 9 MATRIX ALGEBRA AND SIMULTANEOUS LINEAR EQUATIONS Temur Makhkamov Indira Khadjieva

[email protected]
Room IB 205

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

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

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Matrix

A Matrix is simply a rectangular array of numbers arranged in rows

Matrix A Matrix is simply a rectangular array of numbers arranged in
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.

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Examples of matrices

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

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

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Addition (and Subtraction) of matrices

You can add (subtract) two matrices of

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:

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Exercise: Addition and Subtraction

Exercise: Addition and Subtraction

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Scalar multiplication

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

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

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Exercise: Scalar multiplication

Exercise: Scalar multiplication

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Matrix multiplication

Two matrices can be multiplied only if the number of columns

Matrix multiplication Two matrices can be multiplied only if the number of
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:

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Matrix Transpose

The transpose of matrix can be obtained by interchanging the rows

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

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Zero & Identity matrix

Zero matrix is a matrix with all elements 0.
Identity

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

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Determinant of a matrix

A numerical value of matrix
Can be a negative number
Exists

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

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

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

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

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

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Simultaneous linear equations

Simultaneous linear equations

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Equilibrium price in theory

Equilibrium price in theory

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Equilibrium price in practice

Equilibrium price in practice

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Equilibrium point in graph

Equilibrium point in graph

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Breakeven point in theory

Breakeven point in theory

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Breakeven in practice

Breakeven in practice

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Breakeven point in graph

Breakeven point in graph

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Three methods of solving SLE

Three methods of solving SLE

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Gauss’ method (elimination)

Gauss’ method (elimination)

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Matrix inverse method

Matrix inverse method

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Matrix inverse method

Matrix inverse method

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Cramer’s method

Cramer’s method
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