Square Matrix

In linear algebra, a matrix is one of the most widely utilised elements. The rectangular arrangement of numbers/elements/objects is known as a matrix. A matrix’s horizontal arrangement is known as the row, while its vertical configuration is known as the column. The number of rows by columns determines the order of a matrix. 

Square matrix

A square matrix is a matrix with the same number of rows and columns on both sides. The square matrix of order m is known as the m x m matrix in mathematics. The order of the final matrix remains the same when we multiply or add any two square matrices. 

Square matrix of order 2

The square matrix of order 2 is a two-row, two-column matrix. The general form of a 22 matrix, often known as a square matrix of order 2, is: 

Square matrix of order 3

The order of a square matrix of order 3 is 3 since it has 3 rows and 3 columns. The following is the general form of a square matrix of order 3: 

Addition of square matrix

Two square matrices can be added in a very easy fashion, as we described in the introduction section. Consider the 3 by 3 matrix A and B, whose values are as follows: 

For example

As we can see from the above example, adding square matrices is extremely significant. Each row’s value in one matrix is multiplied by the other rows and column’s value in another matrix. 

Multiplication of square matrix

The two square matrices are multiplied in the same manner as the addition method. Consider two 2 by 2 square matrices that must be multiplied. As a result, the final matrix will be: 

Square matrix determinant

A scalar value or a number computed using a square matrix is the determinant of a matrix. The square matrix might have any number of rows and columns, such as 22, 33, or 44, or it could be in the form of n n, with the same number of columns and rows. 

If S is a set of square matrices, R is a set of numbers (real or complex), and f: S R is defined by f (A) = k, where A is a set of square matrices and k is a set of numbers (real or complex), then f (A) is known as the determinant of A. “det A” or | A | denotes the determinant of a square matrix A.

The determinant of a 2 by 2 matrix is given by: 

det A = a₁₁×a₂₂-a₁₂×a₂₁ 

As a result, we may use this formula to get the determinant. 

Properties of square matrix

The following are some of the most important properties of square matrices:

1. There are an equal number of rows and columns. 

2. The trace of a matrix is the sum of all the diagonal components of a square matrix. 

3. An identity matrix is one in which all of the diagonal elements of a square matrix are equal to one. 

4. We can do several operations on a square matrix, such as inverse. 

5. Only the square matrix can be used to calculate the determinant value. 

6. A square matrix’s transpose order is the same as the original matrix’s. 

Important terms relating to square matrix

Order of matrix

In a matrix, it is the sum of the rows and columns. A square matrix has the same number of rows and columns as a rectangular matrix, and the order is n n. 

Trace of a matrix

It equals the sum of a square matrix’s diagonal elements. 

Identity matrix

It’s a square matrix with ones as diagonal elements and zeros as the remaining members. 

Scalar matrix

A square matrix with the same number for all of its diagonal members and 0 for all other elements is known as a scalar matrix. 

Symmetric matrix

A symmetric matrix is one whose transpose is equal to the supplied matrix. 

Skew-symmetric matrix

A skew-symmetric matrix is one whose transpose is equal to the negative of another matrix. 

Orthogonal matrix

If the inverse of a matrix equals the transpose of the matrix, the matrix is said to be orthogonal. 

Conclusion

In the real world, square matrices are used in a variety of ways. Square matrices are invertible and have determinants and can be used to represent and solve systems of equations. Areas and orthogonal vectors can be found using the determinants of square matrices.