Слайд 2
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](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-1.jpg)
tmakhkamov@wiut.uz
Room IB 205
Слайд 3Lecture 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](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-2.jpg)
of matrices to reality
Слайд 4Matrix
A Matrix is simply a rectangular array of numbers arranged in rows
![Matrix A Matrix is simply a rectangular array of numbers arranged in](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-3.jpg)
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.
Слайд 5Examples of matrices
Examples:
1) – 2x2 matrix
2) – 2x3 matrix
3) – ?
![Examples of matrices Examples: 1) – 2x2 matrix 2) – 2x3 matrix 3) – ?](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-4.jpg)
Слайд 6Addition (and Subtraction) of matrices
You can add (subtract) two matrices of
![Addition (and Subtraction) of matrices You can add (subtract) two matrices of](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-5.jpg)
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:
Слайд 7Exercise: Addition and Subtraction
![Exercise: Addition and Subtraction](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-6.jpg)
Слайд 8Scalar multiplication
Multiply each element of the matrix by the number.
Example:
![Scalar multiplication Multiply each element of the matrix by the number. Example:](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-7.jpg)
Слайд 9Exercise: Scalar multiplication
![Exercise: Scalar multiplication](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-8.jpg)
Слайд 10Matrix multiplication
Two matrices can be multiplied only if the number of columns
![Matrix multiplication Two matrices can be multiplied only if the number of](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-9.jpg)
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:
Слайд 11Matrix Transpose
The transpose of matrix can be obtained by interchanging the rows
![Matrix Transpose The transpose of matrix can be obtained by interchanging the](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-10.jpg)
and columns
The first row of the matrix A is the first column of matrix A transposed
Слайд 12Zero & Identity matrix
Zero matrix is a matrix with all elements 0.
Identity
![Zero & Identity matrix Zero matrix is a matrix with all elements](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-11.jpg)
matrix is a square matrix with elements of 1s on the main diagonal from top left to bottom right and 0s on other positions
Слайд 13Determinant 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](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-12.jpg)
for a square matrix only
Determinant for 2x2 matrix is calculated as follows:
Слайд 14Inverse of a (2x2) matrix (1)
If then
![Inverse of a (2x2) matrix (1) If then](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-13.jpg)
Слайд 15Inverse of a (2x2) matrix (2)
Example: , then
Calculate:
Check:
![Inverse of a (2x2) matrix (2) Example: , then Calculate: Check:](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/862041/slide-14.jpg)