1. Matrices and Vector Concepts
Introduction
Vectors
Binary Matrices Operations
Unary Matrix Operations

2. Introduction A matrix is a rectangular array of elements.
Row i of [A] has n elements and is
column j of [A] has m elements and is
The matrix [A] may be denoted by or

3. What are the special types of matrices?
Vector : A vector is a matrix that has only one row or one column
Row Vector: If a matrix [B] has one row, it is called a row vector n is the dimension of the row vector
Column vector: If a matrix [C] has one column, it is called a column vector. m is the dimension of the vector.

4. What are the special types of matrices?
Submatrix: A submatrix of a matrix is obtained by deleting any collection of rows and/or columns, the remaining matrix is called a submatrix of [A] .
Square matrix: the number of rows m of a matrix is equal to the number of columns n. The entries are called the diagonal elements

5. What are the special types of matrices?
Upper triangular matrix:
Lower triangular matrix:

6. What are the special types of matrices?
Diagonal matrix:
Identity matrix:
Zero matrix:

7. Vectors : Introduction
A vector is a collection of numbers in a definite order
If it is a collection of n numbers, it is called a n -dimensional vector

8. Introduction

9. Introduction
A null vector is where all the components of the vector are zero.
A unit vector is defined as
If k is a scalar and is a n -dimensional vector, then

10. Linear Combination of Vectors

11. Linear Combination of Vectors
linearly independent

12. Linear Combination of Vectors

13. The rank of a set of vectors
From a set of n -dimensional vectors, the maximum number of linearly independent vectors in the set is called the rank of the set of vectors. Note that the rank of the vectors can never be greater than the vectors dimension.

14. The rank of a set of vectors
If a set of vectors contains the null vector, the set of vectors is linearly dependent.
If a set of vectors is linearly dependent, then at least one vector can be written as a linear combination of others.

15. Using vectors to write simultaneous linear equations

16. Binary Matrices Operations
Adding matrices
Subtracting matrices

17. Binary Matrices Operations
Matrices multiplication
The scalar product of a constant and a matrix

18. Binary Matrices Operations Rules

19. Binary Matrices Operations Rules
Beware: For multiplication is generally not true

20. Unary Matrix Operations
Transpose Matrix

21. Unary Matrix Operations
Transpose Matrix
Also, note that
Skew-symmetric matrix

22. Unary Matrix Operations
Trace of a matrix

23. Determinant
The determinant of a square matrix is a single unique real number corresponding to a matrix and denotes as or
It is an amazing number containing rich information about a matrix such as inverse and linearly independent vector in matrix
For a 2×2 matrix

24. Properties of Determinant
Determinant of nxn identity matrix is 1
The determinant change sign when two rows are exchanged
The determinant is a linear function of each row separately

25. Properties of Determinant
If two rows of A are equal, then det A = 0
Subtracting a multiple of one row from another row leaves det A unchanged
A matrix with a row of zeros has det A=0

26. Properties of Determinant
If A is a triangular then det A=a11a22…ann=product of diagonal entries
If A is singular then det A =0, if A is invertible then det
The
Question: can all row rules be used for columns as well? Why?

27. The determinant matrix: Cofactor