7. Matrix Inverse

Linear Algebra



Weiqi Zhou

The identity matrix

  • An matrix that is filled with on the main diagonal and elsewhere, denoted by

  • is the multiplicative identity, i.e., holds for any matrix

  • holds for any

The inverse

  • Let be matrices, if , then are inverses of each other, which can be written as and

  • The inverse of a given matrix is unique, a matrix needs to be square to admit an inverse

  • a square matrix is called invertible or non-singular if it admits an inverse, otherwise it is said to be not invertible or singular

Examples



  • is non invertible



Examples



The inverse matrix represents the inverse map

Invertible Bijection Full rank

Inverse of Transpose

thus the inverse of the transpose is usually denoted by

Exercises

  • let be the matrix filled with , is invertible? why?

  • find

  • find

Exercises



  • find

  • find

Solutions to linear systems

  • given with invertible, multiply both sides by we get

  • There are some some rabbits and chicken in the barn. What will be their respective numbers if there are 94 legs and 35 heads in total?

The adjugate

  • let be an matrix, recall that
    the minor of is the determinant of the matrix obtained by deleting the -th row and -th column
    the cofactor of is

  • let be the matrix formed by

  • is called the adjugate of

Adjugate and Inverse

, thus

Verification



  • if , then the RHS is Laplace expansion of by the -th row
    if , say , , then it is the Laplace expansion of by the first row, the determinant is 0 since the first two rows coincide

Example





Exercises

find determinants, adjugates and inverses of the following marices

The Cramer's Rule

  • given with invertible

  • let be the matrix obtained by replacing the -th column of with

  • then

Examples

  • There are some some rabbits and chicken in the barn. What will be their respective numbers if there are 94 legs and 35 heads in total?




Exercises

  • solve following linear systems by using the Cramer's rule



Summary

  • the inverse and itse uniqueness

  • invertible full rank bijection unique solution

  • compute the inverse using the adjugate matrix

  • the Cramer's rule