Table of Contents
What is a Matrix?
A matrix is a rectangular array of numbers, symbols, or other objects. The numbers in a matrix are usually called its elements or entries. The matrix A below has 3 rows and 4 columns.
A =
The matrix above has the following elements:
A[1,1] = 2
A[1,2] = 3
A[1,3] = 4
A[2,1] = 5
A[2,2] = 6
A[2,3] = 7
A[3,1] = 8
A[3,2] = 9
A[3,3] = 10
What are the Different Types of Matrix?
There are different types of matrices, including square matrices, rectangular matrices, and triangular matrices. Square matrices are those that are the same shape on all four sides, rectangular matrices are those that are not square, and triangular matrices have three sides.
Adding Matrices
To add matrices, simply add the corresponding elements of each matrix together.
For example, if A is a 3×2 matrix and B is a 2×3 matrix, then the sum, A+B, would be a 3×5 matrix.
Matrix Sums and Answers
There are a few different ways to find the sum of a series. In this lesson, we will explore two methods: the direct summation method and the partial sum method.
The direct summation method is the simplest method for finding the sum of a series. This method simply involves adding up all of the terms in the series.
The partial sum method is a bit more complex, but it can be more accurate. This method involves breaking the series down into a set of smaller, easier-to-sum series. Then, the sums of the smaller series are added together to find the sum of the original series.
Properties of Matrix Addition
The following are properties of matrix addition:
The sum of two matrices is a matrix that is the sum of the individual matrices.
Matrix addition is associative, meaning that the order of matrix addition does not affect the result.
A matrix is commutative if the order of the matrix’s elements does not affect the result of the matrix addition.
A matrix is distributive if the result of the matrix addition is the same regardless of whether the matrix elements are added or multiplied.
Applications of Matrices
There are a variety of applications in which matrices are used. Some of these applications include:
– solving systems of linear equations
– electrical circuit analysis
– solving problems in physics
– solving problems in engineering
– designing structures
Matrix Types
The different types of matrices are:
| Type of Matrix | Details |
|---|---|
| Row Matrix | A = [aij]1×n |
| Column Matrix | A = [aij]m×1 |
| Zero or Null Matrix | A = [aij]mxn where, aij = 0 |
| Singleton Matrix | A = [aij]mxn where, m = n =1 |
| Horizontal Matrix | [aij]mxn where, n > m |
| Vertical Matrix | [aij]mxn where, m > n |
| Square Matrix | [aij]mxn where, m = n |
| Diagonal Matrix | A = [aij] when i ≠ j |
| Scalar Matrix | A = [aij]mxn where, aij =
where k is a constant. |
| Identity (Unit) Matrix | A = [aij]m×n where, |
| Equal Matrix | A = [aij]mxn and B = [bij]rxs where, aij = bij, m = r, and n = s |
| Triangular Matrices | Can be either upper triangular (aij = 0, when i > j) or lower triangular (aij = 0 when i < j) |
| Singular Matrix | |A| = 0 |
| Non-Singular Matrix | |A| ≠ 0 |
| Symmetric Matrices | A = [aij] where, aij = aji |
| Skew-Symmetric Matrices | A = [aij] where, aij = aji |
| Hermitian Matrix | A = Aθ |
| Skew – Hermitian Matrix | Aθ = -A |
| Orthogonal Matrix | A AT = In = AT A |
| Idempotent Matrix | A2 = A |
| Involuntary Matrix | A2 = I, A-1 = A |
| Nilpotent Matrix | ∃ p ∈ N such that AP = 0 |
