## Matrix in Discrete mathematicsA matrix can be described as a rectangular array of numbers. A matrix that has m numbers of rows and n number of columns, then it will be known as an m∗n matrix. If there is more than one matrix, then it will call matrices. We can indicate the matrix with the help of capital letters. The array of numbers is enclosed with the help of parenthesis or brackets.
We cannot write a matrix by using a straight line instead of brackets. This is because the matrix is used to show a special meaning in the field of linear algebra. If we want to represent the elements, then we can use a shorthand notation, i.e., A = [a
If there is a matrix that has the
Suppose there is a matrix A where In this matrix, the i The (i, j)th element or entry of A can be described as an element a There is a convenient method in which we can simply express the matrix A, i.e., A = [a ## Some Important pointsThere are some terms that we should know when we learn about matrix, which are described as follows:
Each and every value in a matrix will be known as the element, whether it is a number or a constant.
The dimension of a matrix can be described as the number of rows by the number of columns. With the help of dimension, we can name the matrix.
In this matrix, there are 3 number of rows and 2 number of columns. So this matrix has a 3∗2 dimension. In this matrix, there are 2 number of rows and 4 number of columns. So this matrix has a 2∗4 dimension. In this matrix, there are 4 number of rows and 1 number of column. So this matrix has a 4∗1 dimension.
If there is a matrix that has only one row, then it will be a row matrix. This matrix is indicated by [1∗n]. The one-row matrix is described as follows: A[a
If there is a matrix that has only one column, then it will be a column matrix. This matrix is indicated by [m∗1]. The one-column matrix is described as follows:
A matrix will be known as a square matrix if there is an equal number of rows and an equal number of columns. The (n∗n) will be a square matrix.
Suppose there are two matrices A = [a
Now have to determine a, b, c, and d.
A = 1 B = 0 C = -4 D = 2 ## Properties of Matrix- (AB)
^{-1}= B^{-1}A^{-1} - (AT)
^{T}= A and (λA)^{T}= λA^{T} - (A + B)
^{T}= A^{T}+ B^{T} - (A B)
^{T}= B^{T}A^{T}
## Matrix OperationsThere are various operations that can be performed on the matrix, which are described as follows: - Matrix addition
- Matrix subtraction
- Matrix equality
- Matrix multiplication
Now we will explain them one by one.
Suppose there are two matrices, A and B, where A = [a A + B = a Suppose there are two matrices, A and B, which contain the following element: The addition of A and B will be C, which contains the following matrix:
There are some properties that are contained by matrix addition, which are described as follows: Suppose matrix A, B, and C are conformable. Then It will contain the commutative law, which contains the following relation: A+B = B+A It will contain the associative law, which contains the following relation: A+(B+C) = (A+B)+C It will contain the distributive law, which contains the following relation: λ(A+B) = λA + λB, where λ is used to indicate a scalar.
In the subtraction process, we will subtract elements of same position in the given matrices. If there are two matrices, A and B, which have the same dimension or have the same order, only then we can subtract them. If two given matrices have two different orders, then we cannot perform the subtraction operation on them. With the help of symbol A-B, we can indicate the subtraction of A and B. If A = a A - B = a Here i and j can have any value.
Suppose there are two matrices, A and B. These two matrices will be equal if there are the same number of rows and columns in the matrices, and every element at each position in A will have the same element at the corresponding position in B. In this type of matrix, matrices will have the same corresponding entries.
If there are two matrices, A and B, where A has an m∗k matrix and B has a k∗n matrix. We can perform the multiplication operation of two given matrices only if the number of columns in 1 C = AB Here matrix C is used to indicate the resultant matrix. Suppose matrix C contains elements x and y, then it will be defined as C For x = 1....a and y = 1....c In this process, the addition of multiplication of the ith row of matrix A to the corresponding element of jth column of matrix B will be equal to the product of A∗B matrix with its (i, j)th entry.
The dimension of first matrix is 2∗3, and the dimension of the second matrix is 3∗2. So we can see that number of rows in the 1
There are various examples of matrix multiplication, and some of them are described as follows:
Now we have to show whether AB is equal to BA or not.
By multiplying BA, we will get: So we can see that AB This is because matrix multiplication is not commutative.
Now have to determine A(B+C) and AB+AC.
Now we will solve AB+AC like this:
There are some properties that are contained by matrix multiplication, which are described as follows: Suppose matrix A, B, and C are conformable. Then A(B+C)= AB + AC (A+B)C = AC + BC A(BC) = (AB)C ## Identity matrixAn identity matrix can be described as a matrix in which all the diagonal elements of the given matrix are 1. The identity matrix is indicated by the symbol I.
A∗I = A The identity matrix of order n can be described as an n∗n matrix I A
This is a 2∗2 identity matrix because the diagonal element of this matrix 1. This is a 3∗3 identity matrix because the diagonal element of this matrix is also 1. ## Transposes of a matrixSuppose there is a matrix A = [a So transpose of a matrix can be described as a new matrix that can be formed by interchanging the rows and columns. Thus, A is symmetric if and only if A = A'.
Now we have to determine the transpose of this matrix.
A matrix will be known as a null matrix if there is a square matrix in which all the elements are zero. The example of a null matrix is described as follows:
Suppose there is a square matrix A. This matrix will be known as the symmetric matrix if A = A Thus if a
There are some theorems in case of a symmetric matrix, which are described as follows: - If there are two matrices A and B which has n∗n symmetric matrices, then (AB)' = BA.
- If there are two matrices A and B which has n∗n symmetric matrices, then (A+B)' = B+A.
- If there is a matrix C that has any n∗n matrix, then B = C'C will be symmetric.
For example: The following matrix is a symmetric matrix:
A matrix will be known as the zero-one matrix if entries of that matrix are either 0 or 1. We often used this matrix in the form of a table so that we could represent the discrete structures. If we want to define the Boolean operations, we can define their entries in the form of zero one matrix like this: b1 ∧ b2 = 1 if b1 = b2 = 1 0 Otherwise b1 ∨ b2 = 1 if b1 = 1 or b2 = 1 0 otherwise Now we will define Boolean operations on entries in zero-one matrices in table form like this:
Suppose there are two matrices, A and B, which are the m∗n zero one matrices where A = [a
The The Suppose there are two matrices, A and B, where A = [a c With the help of symbol AoB, we can indicate the Boolean product of A and B. Basically, the working of matrices multiplication and Boolean multiplication both are similar, but in
The Boolean multiplication of A and B is described as follows:
Now we have to determine the Boolean product of these matrices.
Suppose there is a zero one matrix A and r is a positive integer. The r A A |