## Matrix
## What is the matrix?The ## Matrix NotationMatrix is usually denoted by a In the above matrix, a The matrix may contain any number of rows and columns. For example: ## Types of MatricesThere are the following types of matrices:
For example:
**Upper Triangular Matrix:**A square matrix in which all the elements below the leading diagonal are zero. In other words, a square matrix**A=[a**is upper triangular if it satisfies the following condition:_{ij}]
a _{ij}=0 for i<jFor example: **Lower Triangular Matrix:**A square matrix in which all the elements above the principal diagonal are zero. In other words, a square matrix**A=[a**is a lower triangular if it satisfies the following condition:_{ij}]
a _{ij}=0 for<ijFor example:
From the above matrix A, we can generate a submatrix. We are deleting the 3. After deleting, we get the following submatrix:^{rd} column## Applications of Matrices**Graph Theory:**The adjacency matrix of a finite graph is a basic notation of graph theory.**Computer Graphics:**In computer graphics, matrices play a vital role in the projection of a three-dimensional image in the 2D screen. Graphics software uses matrix mathematics to process linear transformation to render the image.**Solving Linear Equations:**Using row reduction, Cramer's rule (Determinants), using the inverse matrix.**Robotics:**In robotics and automation, matrices are the base elements of the robot movements.**Recording Experiments:**It is used in many organizations to record the data for their experiments.**Geology:**It is used for seismic surveys.- Cryptography.
## Operations on MatrixWe can perform the following operations on the matrix: - Addition
- Subtraction
- Multiplication
- Division
- Scalar Multiplication
- Inverse
- Transpose
- Negative of a Matrix
## Addition of MatricesThe sum om of two matrices can be done by adding the elements matching with the positions. Remember that both matrices must be of the same size. The resultant matrix is also of the same size. (A+B) _{ij} = A_{ij} + B_{ij}Suppose, there are two matrices A and B, each of size 3×3. The sum of A + B will be: ## Properties of Addition**Commutative Law:**A + B = B + A**Associative Law:**A + (B + C) = (A + B) + C**Additive Identity:**A + 0 = 0 + A = A**Additive Inverse:**A + (-A) = (-A) + A = 0
## Subtraction of MatricesThe subtraction of two matrices can be done by subtracting the elements matching with the positions. In other words, it is an addition of a negative matrix. Remember that both matrices must be of the same size. The resultant matrix is also of the same size. (A-B) _{ij} = A_{ij} - B_{ij}Suppose there are two matrices A and B, each of size 3×3. The subtraction of A - B will be:
## Multiplication of Matrices
First matrix's number of columns = Second matrix's number of rows ## Properties of Multiplication**Non-commutative:**AB ≠ BA**Associative:**A(BC) = (AB)C**Left Distributive:**A(B + C) = AB + AC**Right Distributive:**(A + B)C = AC + BC**Scalar:**k(AB)=(kA)B (where k is scalar)**Identity:**IA=AI=A**Transpose:**(AB)^{T}=A^{T}B^{T}
## Division of MatricesThe division of the matrices is a tricky process. To divide the two matrices, we perform the following steps: - Find the
**inverse**of the**divisor** - Multiply the dividend matrix by the inverse matrix.
Suppose A and B are two matrices, then: Where B
A is the numerator, and B is the denominator. First, we will find the inverse of B. Now multiply the dividend matrix by the inverse. ## Scalar MultiplicationWhen a matrix is multiplied by a Suppose a matrix A of size 3×3 is given. It is multiplied by a constant ## Properties of Scalar MultiplicationLet, A and B two matrices of size m × n, and a and b are two scalars. Then: **Associative Property:**a (b A) = (a b) A**Commutative Property:**aA = Aa**Distributive Property:**(a + b) A = aA + b A and a (A + B) = aA + a B**Identity Property:**1 A = A**Multiplicative Property:**O A = O (where O is a zero matrix)
## The inverse of a MatrixSuppose that we have a square matrix A, whose determinant is not equal to zero, then there exists an m×n matrix AA, where ^{-1} = A^{-1}A = II is the identity matrix.It is easy to find the inverse of a 2×2 matrix in comparison to 3×3 or 4×4 matrix. Follow the steps to find the inverse of a 2×2 matrix. **Swap**the positions of the elements**a**and**d**.- Put a
**negative**sign in front of the**b**and**c** - Divide each element of the matrix by the
**determinant**.
For example, A is a 2×2 matrix. Its determinant is There are three methods to find the inverse of the large matrix. - Gauss-Jordan Method
- Using Adjugate
- Use a matrix calculator
## Properties of Inverse Matrix- A × A
^{-1}= I - A
^{-1 }× A = I - (A
^{-1})^{-1}= A - (A
^{-1})^{T}= (A^{T})^{-1}
## Transpose MatrixWhen we convert the rows into columns and columns into rows and generates a new matrix with this conversion is called the A′, or A or ^{tr},A. For example, consider the following matrix:^{t}The transpose of the above matrix is: ## Properties of Transpose MatrixLet, A and B are two matrices and k is a real number, then: - (A
^{T})^{T}= A - (A + B)
^{T }= A^{T }+ B^{T} - (AB)
^{T }= B^{T}A^{T} - (kA)
^{T }= kA^{T}
## Negative of a MatrixLet, A=a Next TopicMatrix Multiplication |