Learn Linear Algebra With Different Types of Matrices

A Matrix is a rectangular array with a set of entries. It can contain one or more columns and rows. Every entry in the matrix can contain symbols, alphabets, numbers, or others. Entries in the vertical lines are called columns, while those in the horizontal lines are called rows. 

While learning about matrices, there are many types of matrix you must know about.

Rows and Columns in Matrices

Every entry belongs to one column and one row. It is represented as [A]mxn, where the ‘n’ refers to the number of columns and the ‘m’ refers to the number of rows. Hence, an element from the matrix is represented as aji where j and i are the jth row and the ith column where the element belongs. 

The numbers (elements) in a matrix represents data, and sometimes they can even represent mathematical equations. In various time-sensitive engineering, one can multiply matrices for providing good but quick approximations of more complicated calculations.

Besides that, matrices also have real-life applications. They can be used to plot statistics, graphs, and conduct research and scientific studies in almost any field. Matrices can be used for representing real-world data such as infant mortality rates, the population of people, and likewise. 

Moreover, they are also considered the best and most effective representation method to plot surveys. 

Types of Matrices As Per Dimensions

Although matrices are of varied sizes, their shapes usually remain the same. The dimension or size of the matrix is the total number of columns and rows in a matrix. Hence, here are the types of matrices depending on dimension. 

1. Column and Row Matrix

Row matrices have several columns but only one row. On the contrary, column matrices have only one column but multiple rows.

2. Square and Rectangular Matrix

A matrix that has an unequal number of columns and rows is a rectangular matrix. Such rectangular matrices are represented as [B]mxn. Similarly, if a matrix has an equal number of columns and rows, it is called a square matrix. The square matrix is represented as [B]nxn. 

3. Constant Matrices

In these matrices, all elements are constants irrespective of the size or dimension of the matrix. The elements of the matrix are represented as bij. Here are some examples of matrices with constant elements:

• Identity Matrix: it is a square diagonal matrix where all entries located diagonally are equal to 1 while the rest elements are O. This matrix is represented as I. 
• Matrix of Ones: a matrix where all elements are equal to 1 is referred to as the matrix of ones. 
• Zero Matrix: Here, all elements are equal to 0. 

Other Kinds of Matrices

Besides these common types of matrices, here are some other kinds of matrices used in computer technologies and advanced mathematics. Here are some of them:

1. Non-singular and single Matrix

A non-single matrix has a determinant not equal to zero, whereas a singular matrix has a determinant equal to 0. One can use the determinant formula for finding a matrix’s determinant. 

2. Lower and Upper Triangular Matrix

In a lower triangular matrix (square-shaped), all elements present above the diagonal are 0. But in an upper triangular matrix (square-shaped), all elements below the diagonal are 0. 

3. Skew Symmetric and Symmetric Matrix

A skew-symmetric matrix is one where a square matrix G of nxn size is GT= -G, and a symmetric matrix is one where a square matrix B of nxn size is BT=D. 

4. Boolean Matrix

A boolean matrix has all its elements as either 0s or 1s.

5. Stochastic Matrix

In this matrix, all elements represent probability. For instance, a square matrix A will be left stochastic if all its elements are non-negative and the elements in every column sum up to 1. However, a matrix with all elements as non-negative and the elements in every row sum up to 1 is a right stochastic matrix.

6. Orthogonal Matrix

An orthogonal matrix is one when AxAT=I in a square matrix A. Here, BT  is matrix B’s transpose, and I is an identity matrix. 

Conclusion

Now you know the numerous types of matrices and how each matrix is used in mathematics and practical situations through real-life application.

FAQs

1. What are matrices and their different types? 

Answer: A matrix is an array of numbers that consists of columns and rows. These columns and rows define the dimension or size of a matrix. There are various types of matrices such as antisymmetric matrix, symmetric matrix, lower triangular matrix, upper triangular matrix, diagonal matrix, square matrix, column matrix, row matrix, and null matrix. 

2. What is a scalar matrix? 

Answer: Both square matrix and scalar matrix have similar properties. All off-diagonal elements are equal to 0s as well as non-diagonal elements are equal. Hence, it can be concluded that a scalar matrix is the multiple of an identity matrix. 

3. Is a zero matrix invertible?

Answer: A zero matrix will not be invertible as its determinant is zero.