Introduction to Matrix Inverse

The inverse of the matrix \(A\) is denoted by \(A^{-1}\). In this step-by-step guide, you learn more about the formula, methods, and terms related to the inverse of a matrix.

Introduction to Matrix Inverse

A matrix is a specific set of objects arranged in rows and columns. These objects are called matrix elements. The inverse matrix can only be found for square matrices whose number of rows and columns is equal, such as \(2 × 2\), \(3 × 3\).

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Step by step guide to an introduction to matrix inverse

The inverse of a matrix is another matrix that, when multiplication with a given matrix, gives a multiplicative identity.  For a matrix \(A\), its inverse is \(A^{-1}\) and \(A.A^{-1}=A^{-1}.A= I\), where \(I\) is the identity matrix.

An identity matrix is a square matrix that has ones on its diagonal and zeros everywhere else. Consider the identity matrix as the number \(1\) in the matrix world.

The invertible matrix is a matrix whose determinant is non-zero and for which the inverse matrix can be calculated.

Inverse matrix formula:

\(\color{blue}{A^{-1}=\frac{1}{|A|}. Adj A}\)

where \(A\) is a square matrix.

Note: For inverse of a matrix to exist:

  • The given matrix should be a square matrix.
  • The determinant of the matrix should not be equal to zero (\(|A| ≠ 0)\).

Terms related to inverse of matrix:

  • Minor:

Minor is defined for every element of a matrix. The minor of a particular element is the determinant obtained after removing the row and column containing this element.

  • Cofactor:

The cofactor of an element is calculated by multiplying the minor with -1 to the exponent of the sum of the row and column elements to display that element.

Cofactor of \(a_{ij}=(-1)^{i+j} \times minor \:of\: a_{ij}\)

  • Determinant:

The determinant of the matrix is equal to the summation of the product of the elements and their cofactors, of a particular row or column of the matrix.

  • Singular Matrix: 

A matrix with a determinant value of zero is known as a singular matrix. For a single matrix \(A\), \(|A| = 0\). The inverse of a singular matrix does not exist.

  • Non-Singular Matrix:

A matrix whose determinant value is not equal to zero is called a non-singular matrix. For a non-singular matrix \(|A| ≠ 0\). A non-singular matrix is called an invertible matrix because its inverse is computable.

  • Adjoint of Matrix:

The adjoint of a matrix is the transpose of the cofactor element matrix of the given matrix.

Methods to find the inverse of a matrix:

The inverse of the matrix can be found using two methods:

1. The elementary operation: the elementary operations on a matrix can be performed through row or column transformations:

Elementary row operations:

To calculate the inverse of matrix \(A\) using elementary row transformations, first, take the augmented matrix \([A | I]\), where \(I\) is the identity matrix whose order is like \(A\). Then apply the row operations to convert the left side \(A\) into \(I\). Then the matrix gets converted into \([I | A^{-1}]\).

Elementary column operations:

Column operations can also be applied, such as how to explain the process for row operations to find the inverse of a matrix.

2. The use of an adjoint matrix: the inverse of a matrix can be calculated by applying the inverse formula of the matrix using the determinant and joining the matrix. The inverse of a matrix can be calculated by following the given steps:

Step 1: Calculate the minor of all elements \(A\).

Step 2: Then calculate the cofactors of all the elements and write the cofactor matrix by replacing the elements \(A\) with the corresponding cofactors.

Step 3: Find the adjoint of \(A\) by taking the transpose of the cofactor matrix of \(A\).

Step 4: Multiply adj \(A\) by reciprocal of determinant.

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