Linear algebra mainly focuses on the study of straight lines, planes, vectors, and spaces. The equations that are formed using linear algebra will always have the highest power >1.
Here is 3x, 3 is the coefficient, and x is the variable or the unknown, again in 4y + 5, 4 and 5 are the constants, and y is the unknown.
The linear algebraic equations form a system that can be solved with the help of a matrix. Before we start solving the equations, we need to know about different types of matrices. Let us know about different Matrix Types.
What is a Matrix?
A Matrix refers to the organized rectangular arrangement or rectangular table or array of numbers that can be whole or real or complex or even functions. The Matrix is written within [ ] or ( ). For example:
Order of a Matrix
The order or size of the matrix can be determined by the number of rows along with the number of columns present. Like, if a matrix has n number of rows, and m number of columns, then the order of the matrix will be n x m.
Different types of Matrices: Details
Matrices consist of various types with various functions, which help solve the equations easily with a basic knowledge of algebra. They are as follows:
● Row matrix
A matrix that consists of only one row is known as a Row Matrix. Hence, if A= n x m, and n = 1only, the matrix becomes a row matrix. As there is only one-row present, so, the order becomes 1 x m. example of Row Matrix:
A= [ 1 2 3 ], C = [ 5 6 7 9 10]
● Column matrix
A matrix that consists of only one column is known as the Column Matrix. Hence, if A = n x m and m = 1, the order of the matrix becomes n x 1. Hence the column matrix is formed. Example of Column Matrix:
● Null matrix
If all the elements in a matrix are zero, the matrix is known as the null matrix or zero matrices. It is generally represented by 0. Hence, A= [ aij] n x m is 0 matrix if aij=0 for all i and j. Examples of Null or Zero Matrix:
● Square matrix
A matrix that consists of an equal number of rows and an equal number of columns, then the matrix is known as a square matrix. Hence, A= [ aij] n x m will be a square matrix if n = m. For example
● Diagonal matrix
If all the elements p[reent in the matrix, except the main diagonal, in a square matrix, are zero, it is known as Diagonal Matrix. Thus, if A= [ aij] will become a diagonal matrix, if only aij = 0 but i is not equal to j. Here, the diagonal matrix is in the order of 3 x 3.
Hence it can be also written as [ 1 1 1 ]. The main thing that needs to be observed is that the diagonal elements in the matrix are non-zero, and the rest of the elements present are zero. The two things about the diagonal matrix that you should always remember are:
- A diagonal matrix will always be a square matrix.
- A diagonal matrix will only have one diagonal present, and the rest of the elements are characterized by the general form: aij, where i = j.
Examples of Diagonal Matrix:
● Triangular matrix
A matrix consisting of elements present in the square matrix whose above and below diagonals are zero is a Triangular Matrix. Triangular Matrices consist of two types: Upper Triangular Matrix and Lower Triangular Matrix.
Upper Triangular Matrices:
A square matrix can only be known as the Upper Triangular Matrix if aij = 0, where i > j. Example
Lower Triangular Matrices:
A square matrix can only be known as the Lower Triangular Matrix if aij = 0, where i < j. Example
● Scalar matrix
A matrix in which all the elements in the diagonal of a diagonal matrix are equal is known as the Scalar Matrix. Hence, a square matrix, A= [ aij], can be a scalar matrix only if the diagonal elements are equal, i= j. Example:
● Identity or unit matrix
In a matrix, if all the main diagonal elements of a diagonal matrix are 1, then the matrix is known as the Identity Matrix or unit Matrix. A unit matrix of an order m is denoted as I’m. Hence a square matrix A= [ aij]n x m is an identity matrix if aij = 1, i = j and aij = 0 if i ≠ j. Example of the Identity matrix:
Hence, while solving the identity matrix, you will observe three things:
- An Identity Matrix will always be a square matrix.
- An Identity Matrix will always be a diagonal matrix.
- An Identity Matrix will always be a Scalar matrix.
● Symmetric matrix
In a square matrix A= [ aij]n x m, if aij = aji, for every value of i and j, then the matrix is known as a Symmetric Matrix. Example of Symmetric Matrix
Here, a12 = a21 = 2 and a13 = a31= 3
● Skew-symmetric matrix
In a square matrix A= [ aij]n x m, if aij = aji, for every value of i and j, it is known as a skew-symmetric matrix. But, in a skew-symmetric matrix, all diagonal elements will be equal to zero, i.e. aii = 0. Example of Skew Symmetric Matrix
We hope this article has helped you in understanding the different types of matrices. Practice each of them to effortlessly ace this topic in maths.
1. Who is known as the father of Matrices?
Arthur Cayley, an English lawyer, and mathematician was the first person to establish the matrix method. He is known as the father of Matrices.
2. What are the different applications of matrices?
The matrix method is used to plot graphs, used in scientific research, or even in statistics. Many of the designing software run with the application of matrix methods.
3. In which field can the matrices be applied?
Matrices are applied in different fields of engineering, scientific research, statistical fields, etc. Matrices are also used to build many computer applications and software. Many surveyors also use matrices to produce maps and show uneven surfaces.
Read More: How To Remember Maths Formulas In Class 8th.