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Resources on Linear Algebra

Matrix Inverse

By Kardi Teknomo, PhD.
LinearAlgebra

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When we are dealing with ordinary number, when we saymatrix inverse then we can obtainmatrix inverseas long asmatrix inverse. We write it asmatrix inverseormatrix inverse. For example, the inverse of 2 is ½. Matrix inverse is similar to division operation in ordinary numbers. Suppose we have matrix multiplication such that the result of matrix product is an identity matrix matrix inverse. If such matrix matrix inverseexists, then that matrix is unique and we can write matrix inverse or we can also writematrix inverse.

Matrix inverse exists only for a square matrix (that is a matrix that has the same number of rows and columns). Unfortunately, matrix inverse does not always exist. Thus, we give name that a square matrix is singular if that matrix does not have an inverse matrix (remember a single person does not have a spouse). When a square matrix has an inverse, it is called non-singular matrix.

Because matrix inverse is a very important operation, in linear algebra, there are many ways to compute matrix inverse.

  1. The simplest way to find matrix inverse for a small matrix (order 2 or 3) is to use Cramer’s rule that employ determinant of the input matrix. Recalled that a square matrix is singular (i.e. no inverse) if and only if the determinant is zero. Matrix inverse can be computed by scaling the adjoint of the input matrix with the determinant, that ismatrix inverse. Adjoint of the input matrix is a matrix whose element is the cofactor of the input matrix.
    For a 2 by 2 matrix, we can even memorize the formula
    When matrix inversethen matrix inverse
    The determinant is set by multiplying the diagonal elements minus the product of the off-diagonal elements and the adjoint is set by reversing the diagonal elements and taking the minus sign of the off diagonal elements.
  2. For matrix of medium size (order 4 to 10), the usage of elementary row operations to produce RREF matrix using Gaussian Elimination or Gauss Jordan is useful.
  3. Other techniques to compute matrix inverse of medium to large size are to use numerical methods such as LU decomposition, SVD, or Monte Carlo method.
  4. For a large square matrix (order more than 100), numerical techniques such as Gauss-Siedel or Jacobi method are used to approximate the matrix inverse.

Example
Find matrix inverse ofmatrix inverse
Using determinant matrix inverse
We havematrix inverse

The interactive program below is using numerical methods. As this is an educational program, I limit the matrix size to square matrix of medium size up to order 10. Random Example button will create new random input matrix.


Report in rational format

Properties

Some important properties of matrix inverse are

  • If matrix inverseand matrix inverseare square matrices order matrix inverseand their product produce an identity matrixmatrix inverse, then matrixmatrix inverse.
  • If a square matrix matrix inversehas an inverse (nonsingular), then the inverse matrix is unique. There is no other inverse matrix.
  • A square matrix matrix inversehas an inverse matrix if and only if the determinant is not zeromatrix inverse. Similarly, matrix matrix inverseis singular (has no inverse) if and only if the determinant is zeromatrix inverse.
  • A square matrix matrix inverseorder matrix inversehas an inverse matrix if and only if the rank matrix is full, that ismatrix inverse.
  • If a square matrix matrix inversehas an inverse, the determinant of an inverse matrix is the reciprocal of the matrix determinant, that is matrix inverse.
  • If a square matrix matrix inversehas an inverse, and we have a scalar matrix inversethen the inverse of a scalar multiple is equal to product of their inverses, that ismatrix inverse.
  • If a square matrix matrix inversehas an inverse, the transpose of an inverse matrix is equal to the inverse of the transpose matrix, that ismatrix inverse.
  • If matrix inverseand matrix inverseare square nonsingular matrices order matrix inversethen the inverse of their product is equal to the product of their inverse in reverse order, that ismatrix inverse .
  • Let matrix inverseand matrix inverseare square matrices ordermatrix inverse. If matrix inversethen either matrix inverseor matrix inverse or both matrix inverse and matrix inverseare singular matrices (no inverse).

See also: matrix multiplication, matrix transpose, determinant, rank

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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