Square Matrix

Square Matrix: A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. A square matrix is characterized by having the same number of rows as columns. It's referred to as an m x m matrix, indicating that it is a square matrix of order ?m. When two square matrices are either multiplied or added, the resultant matrix retains the same order. For example, consider the square matrix X given by

X = \begin{bmatrix} 2 & -7 & 7 \\ 2 & 5 & 6 \\ 5 & 4 & 3 \end{bmatrix}

In this article, we will learn about what is a square matrix, along with its examples, properties, types, formula, and some problems on square matrix.

What is a Square Matrix?

A square matrix is defined as a matrix that has an equal number of rows and columns. The order of a square matrix that has "n" rows and "n" columns is "n × n." The number of elements in a matrix can be determined by the product of the number of rows and columns in the matrix. So, the number of elements in a square matrix is always a perfect square number. If any two square matrices of the same order are added or multiplied, the order of the resultant matrix remains the same.

Square Matrix Definition

A square matrix is a matrix with the same number of rows and columns. It is an n x n matrix, where n represents the number of rows or columns.

Square Matrix Examples

Some common examples of square matrices of different orders are given below:

• Following matrix is a square matrix of order "2 × 2," i.e., the given matrix has two rows and two columns.

 A_{2\times2} = \left[\begin{array}{cc} p & q\\ r & s \end{array}\right]

• Following matrix is a square matrix of order "3 × 3," i.e., the given matrix has three rows and three columns.

B_{3\times3} = \left[\begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{array}\right]

• Following matrix is a square matrix of order "n × n," i.e., the given matrix has "n" rows and "n" columns.

A_{n\times n} = \left[\begin{array}{cccccc} a_{11} & a_{12} & a_{13} & . & . & a_{1n}\\ a_{21} & a_{22} & a_{23} & . & . & a_{2n}\\ a_{31} & a_{32} & a_{33} & . & . & a_{3n}\\ . & . & . & . & . & .\\ . & . & . & . & . & .\\ a_{n1} & a_{n2} & a_{n3} & . & . & a_{nn} \end{array}\right]

Properties of a Square Matrix

The following are some important properties of a square matrix:

• In every square matrix, the number of rows and columns is equal.

• The sum of all principal diagonal elements in a square matrix is defined as the trace of a matrix.

• The determinant value can be determined only for square matrices.

• If the determinant of a square matrix is zero, then the matrix is called a singular matrix, whereas it is said to be non-singular if its determinant is not equal to zero.

• If the principal diagonal elements in a square matrix are ones and the rest of the elements are zeros, then the matrix is called an identity matrix.

• The order of a square matrix and its transpose are the same.

• We can perform various operations on a square matrix, such as addition, multiplication, inverse, etc.

Types of Square Matrix

We have various types of square matrices, such as identity matrices, singular matrices, non-singular matrices, symmetric matrices, etc.

• Singular Matrix: If the determinant of a square matrix is zero, then the matrix is called a singular matrix.

"A" is a square matrix of order "n × n" and is said to be a singular matrix if |A| = 0.

• Scalar Matrix: A scalar matrix is a square matrix whose principal diagonal elements are equal and the rest of the elements are zeros.
• Identity Matrix: An identity matrix is a square matrix whose principal diagonal elements are ones and the rest of the elements are zeros.
• Symmetric Matrix: A symmetric matrix is a square matrix whose transpose is the same as the original matrix.

"A" is a square matrix of order "n × n" and is said to be a symmetric matrix if A^T = A. 

• Skew-Symmetric Matrix: A skew-symmetric matrix is a square matrix whose transpose is equal to the negative of the original matrix.

"A" is a square matrix of order "n × n" and is said to be a symmetric matrix if A^T = −A. 

• Orthogonal Matrix: An orthogonal matrix is a square matrix whose inverse is equal to the transpose of the matrix.

"A" is a square matrix of order "n × n" and is said to be an orthogonal matrix if A^-1 = A^T.

Addition and Subtraction of Square Matrices

We can perform various mathematical operations on a square matrix, such as addition, multiplication, etc. When two square matrices of the same order are added or subtracted, the resultant matrix is obtained when corresponding elements are added or subtracted.

If A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] and B = \left[\begin{array}{cc} p & q\\ r & s \end{array}\right]              are two square matrices of order "2 × 2," then A\pm B = \left[\begin{array}{cc} a\pm p & b\pm q\\ c\pm r & d\pm s \end{array}\right]  .

• If A and B are two square matrices of the same order, then A + B = B + A.

Multiplication of Square Matrices

If are A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] and B = \left[\begin{array}{cc} p & q\\ r & s \end{array}\right]  are two square matrices of order "2 × 2," then 

A\times B = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] \times \left[\begin{array}{cc} p & q\\ r & s \end{array}\right]

A\times B = \left[\begin{array}{cc} (a\times p)+(b\times q) & (a\times r)+(b\times s)\\ (c\times p)+(d\times q) & (c\times r)+(d\times s) \end{array}\right]              .

Determinant of a Square Matrix

The determinant of a square matrix is a scalar value or a summary value that represents the entire set of elements in the matrix. The determinant of a "2 × 2" square matrix can be easily determined using the formula, which is the difference of the product of the matrix's principal diagonal elements.

If A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]   is a square matrix of order "2 × 2," then the formula to calculate its determinant is given as follows:

|A| or det A = [ad – bc]

Inverse of a Square Matrix

Inverse operation can be performed on the square matrix. Division of a matrix can be achieved using the inverse of a Matrix. Inverse of a matrix is found after the determinant of the Square Matrix is calculated. After that Adjoint of the Square matrix is found. Then Inverse is calculated as,

A^-1= adj. A / |A|

Articles Related to Square Matrix:

• Minors and Cofactors of Determinants
• Triangular Matrix
• Diagonal Matrix

Solved Examples on Square Matrix

Example 1: Calculate the determinant of the square matrix given below.

P = \left[\begin{array}{cc} -3 & 5\\ -2 & 0 \end{array}\right]

Solution:

Given matrix, P = \left[\begin{array}{cc} -3 & 5\\ -2 & 0 \end{array}\right]

|P| = (-3 × 0) – (–2 × 5)

|P| = 0 – (–10) = 10

Hence, the determinant of the given matrix is 10.

Example 2: Find whether the matrix given below is a singular matrix or not.

B = \left[\begin{array}{cc} 0 & 4\\ 4 & 0 \end{array}\right]

Solution:

To find whether the matrix given below is singular or not, we have to calculate its determinant.

|B| = (4 × 4) – (0 × 0) = 16 ≠ 0

As the determinant of the given matrix is not equal to zero, the given matrix is non-singular.

Example 3: Determine the inverse of the matrix given below.

A = \left[\begin{array}{cc} 1 & -2\\ 6 & 7 \end{array}\right]

Solution:

If A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]           , then

A^{-1} = \frac{1}{ad-bc} \left[\begin{array}{cc} d & -b\\ -c & a \end{array}\right]

A^{-1} = \frac{1}{7+12} \left[\begin{array}{cc} 7 & 2\\ -6 & 1 \end{array}\right]

A^{-1} = \frac{1}{19}\left[\begin{array}{cc} 7 & 2\\ -6 & 1 \end{array}\right]

A^{-1}= \left[\begin{array}{cc} \frac{7}{19} & \frac{2}{19}\\ \frac{-6}{19} & \frac{1}{19} \end{array}\right]

Example 4: Find whether the matrix given below is symmetric or not.

A = \left[\begin{array}{ccc} 2 & 0 & 6\\ 5 & 4 & 9\\ 1 & 3 & -5 \end{array}\right]

Solution:

"A" is a square matrix of order "n × n" and is said to be a symmetric matrix if A^T = A.

The given matrix, A = \left[\begin{array}{ccc} 2 & 0 & 6\\ 5 & 4 & 9\\ 1 & 3 & -5 \end{array}\right]

A^{T} = \left[\begin{array}{ccc} 2 & 5 & 1\\ 0 & 4 & 3\\ 6 & 9 & -5 \end{array}\right]           ≠ A

Since, AT ≠ A, the given square matrix is not symmetric.