Involuntary Matrix

Matrix or matrices (plural) are a simple array of numbers organised as rows and columns. The concept of matrices is extremely important in the field of linear algebra. This tool alone significantly reduces the workload by a huge margin as compared to other straightforward concepts in the field of mathematics.

The applications of matrices are of great significance as they cover a wide field of topics. We use them in different areas of business and science. When performing functions of financial management and statistics like budgeting, analysing the results of an experiment, etc matrices prove to ‌be very helpful.

What are Matrices?

Theoretically, matrices are an ordered rectangular array of numbers or functions. These numbers or functions are called entries or the elements of a matrix.

Imagine a situation where we wish to express that Rob has 10 notebooks. We can write  it as [ 10 ]. Here the number inside  [ ] denotes the number of notebooks Rob has. Now if we are in a situation where Rob has 10 notebooks and 5 pencils. We can write it as [10 5]. Here the first number represents the number of notebooks and the second number represents the number of pencils owned by Rob.

This information was easy to express since we took into consideration only one person. Now suppose we want to express this information:

We can express this information in the following way using matrices:

This method of organising numbers or functions into an array of rows and columns forms a matrix and plural of the matrix is matrices. The order of the matrices is determined by the number of rows and columns it has. The matrix above has 3 rows and 2 columns. We show the rows and columns as m X n. With the above matrix, we can represent it as a 3 X 2 matrix.

Types of Matrices

A major part of linear algebra lies in the concepts of vectors and matrices. There are a few types of matrices that are quite common in these concepts. We have discussed below, the 6 main types of matrices that are extremely important in order to understand the topic.

Square Matrix

We defined square matrices as the matrices which have the same number of rows (m) and the same number of columns (n). The name comes from the shape this matrix forms in comparison with the more traditional rectangular matrix.

Here, the number of rows (m) is equal to the number of columns (n) which is 3 X 3. This is an example of an order 3 matrix.

Symmetric Matrix

A symmetric matrix is a matrix in which the top left triangle is identical to the bottom right triangle of the matrix. It is one of the most important types of matrices and is crucial for solving the questions of linear algebra and matrices

Given above, is an example of a matrix with 4 rows (m) and 4 columns (n) where the top left triangle is identical to the bottom right triangle of the matrix. Hence, this is called a square symmetric matrix of order 4.

Triangle Matrix

A triangle matrix is one that has all of its values in the top right or the bottom left triangle. The rest of the values or elements it contains are 0. If the values in a triangle matrix are in its upper part, it is called an upper right triangular matrix. However, if the values in the matrix are in the bottom-left part of the matrix, it is called a bottom left triangular matrix.

Given above is an example of an upper triangular matrix of order 3 since it has 3 rows and 3 columns.

Involutory Matrices

An involuntary matrix is a uniques matrix. For a matrix Z for it to be involutory. Z2 =  I where I is an identity matrix. Therefore, for a matrix to be involutory, its square should be equal to the identity matrix of the same order.

An involutory matrix is its own inverse as it gives an identity matrix when multiplied by itself. Therefore it can be said that all involutory matrices are square roots of I given they are of the same order.

Example of Involutory Matrix

Z =

Z2 =  

Properties of Involutory Matrix

After understanding the definition and example of involutory matrices, we can now go ahead and discover the properties involutory matrices show:

• Given that A and B are involutory matrices and they satisfy the condition AB = BA. Then, AB will also be an involutory matrix.

• Involutory matrices always have +1 and -1 as their Eigenvalues. Eigenvalues are a scalar quantity associated with most matrix equations.

• A diagonal matrix derived from an involutory matrix will also be an involutory matrix.

• If an involutory matrix satisfies the condition of being an identity matrix, it is also an idempotent matrix.

Conclusion

This article covers what a matrix is, applications of a matrix in various fields and how it comprises a major part of linear algebra. We also discussed what an involutory matrix is and its properties. One must remember that a matrix Z is involutory only if 

• Z = Z-1

Z2 = I where I is an identity matrix.