Square Matrix

The matrices are the arrangement of complex and real numbers, symbols and expressions in a tabular form containing rows and columns. Hence, the number of rows and columns of a matrix is an important matrix property. The number of rows and columns present in a matrix is represented by the order of a matrix. If, in a particular matrix A, the number of columns is equal to the number of rows present, then matrix A is said to be a square matrix. Let us study the properties and examples of a square matrix.

Types of matrices based on the order

The order of the matrix is the representation of the number of rows and the number of columns present in a matrix. The order is represented in terms of m and n, where m is the number of rows present in a matrix and is the number of columns present.

The matrices can be of any order. However, based on their order, the matrices are generally of two types, as given below.

• Rectangular matrix:

The number of rows and the number of columns in a rectangular matrix is not equal. Hence the order of a rectangular matrix is m×n, where m and n both have distinct values.

• Square matrix:

The number of rows in a square matrix equals the number of columns in the same square matrix. Let us study in-depth about this square matrix.

• Square matrix:

The square matrix definition is as given below.

A square matrix is a type of matrix in which the number of columns equals the number of rows. The order of a square matrix can be represented with n×n, and n represents the value of a number of rows as well as the number of columns. 

The total number of elements in a matrix is equal to the number of rows multiplied by the number of columns. So, in a square matrix, we are multiplying a number by itself, so the number of elements in a square matrix is always a perfect square.

A square matrix example is given below.

The matrix shown above is of the order 2×2. Since the number of rows is equal to the number of columns of the matrix, the above matrix is a square matrix.

The value of n in this matrix will be equal to 2. Hence, the total number of elements in the above square Matrix will be equal to 2², that is 4.

Let us study a few properties of a square matrix.

Properties of a square matrix

Given below are a few properties of a square matrix.

• The number of columns in a square matrix equals the number of rows of that matrix.

• The order of a square matrix is represented as n×n.

• The total number of elements in a square matrix is always a perfect square.

• When a square matrix is transposed, the order of the transposition of the matrix is the same as the original matrix.

• A square matrix has a determinant value.

Let us study a few square matrices.

Types of the square matrix

Square matrices are further classified based on their properties. A few types of square matrices are stated below.

• Identity matrix:

An identity matrix is a type of square matrix in which the diagonal elements are all equal to one. In contrast, the rest of the elements are equal to zero.

• Symmetric matrix:

If a transpose of a matrix is equal to the original matrix and no elements are changed when the transpose is obtained, then that matrix is called a symmetric matrix.

• Skew-symmetric matrix:

If a transpose of a matrix A is equal to the negative of the same matrix, then matrix A will be classified as a Skew-symmetric matrix. 

• Scalar matrix:

Suppose in a square matrix, and all the diagonal elements are equal; that is, the same number of symbols is used to fill all the diagonal positions, and the rest of all elements of the matrix are equal to zero, then the matrix is known as the scalar matrix.

• Orthogonal matrix:

Suppose there is a matrix A. If the inverse of the matrix A is equivalent to the transpose of the same matrix A, the. Matrix A is known as the orthogonal matrix.

Such are the types of square matrices.