Inverse of matrix by elementary operations

The row reduction method, commonly known as the Gaussian Elimination method, is an approach for solving a system of linear equations. It’s commonly thought of as a series of operations on the corresponding coefficient matrix.

Elementary operations and Types

A matrix is a collection of numbers organized into rows and columns. The dimensions of a matrix are the number of rows and columns, which are denoted as m n, where m and n are the number of rows and columns, respectively. There are certain basic operations that can be performed on a matrix aside from basic mathematical operations. The operations done on rows and columns of a matrix to transform the given matrix into a new form in order to make the calculation simpler are known as elementary operations or transformations of a matrix.

Elementary Row Operations: Primary or elementary row operations are operations that are done on the rows of an array or matrix.

Elementary Column Operations: Primary or elementary column operations are the basic matrix operations done on its columns.

A matrix’s three basic elementary operations or transformations are as follows:

• Any two rows or columns can be changed.

• Multiplication of a non-zero value by a row or column.

• Add the result to the other row or column after multiplying the row or column by a non-zero value.

Operations Used on Matrix to modify

The following operations on the matrix can be used to modify it.

• Two rows are interchanged/swapped.

• A positive integer is multiplied or divided by a row.

• A multiple of one row is added or subtracted from another.

We may now change a matrix and find its inverse using these techniques. The steps are as follows:

Step 1: Make an n x n identity matrix.

Step 2: To make the original matrix (A) equivalent to the identity matrix, perform row or column operations on it.

Step 3: Apply the same techniques to the identity matrix.

Formula for an Inverse Matrix

To divide one matrix by another matrix, the inverse of a square matrix is employed. To find the inverse of a matrix, we must first determine the matrix’s determinant. The square matrix’s adjoint is then calculated. Finally, the inverse is calculated by dividing the adjoint matrix by the square matrix’s determinant.

Suppose, the inverse of any real integer A^-1 . At times A^-1 , equaled 1. We understood that the reciprocal of a real number was the inverse of the number as long as the number was not zero. The identity matrix is the product of A and A^-1  which is the inverse of A, indicated by  A^-1. The resulting identity matrix will be the same size as the matrix.

    A^-1=1/ |A|. Adj A

The inverse of a matrix exists only if the determinant of the matrix is a non-zero number, because |A| is in the denominator of the expression. In other words, |A| ≠ 0.

Square matrices

A square matrix is a matrix having the same number of rows and columns. A square matrix of order n is defined as an n-by-n matrix. The addition and multiplication of any two square matrices of the same order is possible.

Simple linear transformations, such as shearing or rotation, are frequently represented by square matrices.

A square matrix has the same number of rows and columns. It has the shape n x n in its order. In addition, the square matrix’s number of elements is determined by the product of these rows and columns. As a result, the total number of elements is always a perfect square number.

 The following is an example of a square matrix.

The number of rows of the given matrix is 3.

The number of columns of the given matrix is 3.

The given matrix A is a square matrix because the number of rows and columns are equal.

Inverse of a square matrix

To divide one matrix by another matrix, the inverse of a square matrix is employed. To find the inverse of a matrix, we must first determine the matrix’s determinant. The square matrix’s adjoint is then calculated. Finally, the inverse is calculated by dividing the adjoint matrix by the square matrix’s determinant.

Let’s take a look at an important term related to a matrix’s inverse. If the transpose of a square matrix equals its inverse, it is called an orthogonal matrix.

Orthogonal Matrix: A^t=A^-1

Conclusion

The row reduction method, commonly known as the Gaussian Elimination method, is an approach for solving a system of linear equations.

The inverse of a matrix exists only if the determinant of the matrix is a non-zero number, because |A| is in the denominator of the expression.