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Adjoint and Inverse of a Matrix

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Adjoint and Inverse of a Matrix - Lesson Summary

The inverse of a square matrix obtaining by using elementary row or column operations.


Adjoint of a matrix

Consider A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

Let Aij = Cofactor of aij    a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33    ↔    A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33    


Transpose of matrix   A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33   =   A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33  

According to the definition, adj A =   A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33  

Adjoint of a matrix

Let A = a b c d

The cofactors of the elements of A are

A11 = d, A12 = -c, A21 = -b, A22 = a

A =  A 11 A 21 A 12 A 22 = d –b –c a


Theorem 1

If A is any given square matrix of order n, then:

A(adj A) = (adj A) A = |A| I, where I is the identity matrix of order n.


Singular matrix:

A square matrix is said to be singular if its determinant is zero.


Non-singular matrix:

A square matrix is said to be non-singular if its determinant is not zero.


Theorem 2

If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.

Claim: |AB| = |A||B|


Theorem 3

The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B| , where A and B are square matrices of the same order.


Theorem 4

If A is a square matrix of order n, then |(Adj A)| = |A|n-1.


Inverse of a square matrix:

A square matrix, A, is said to be an invertible matrix if there exists a square matrix, B, such that AB = BA = I.

If a square matrix, A, is invertible, then it is a non-singular matrix.


Theorem 5

If A is a non-singular matrix, then A is invertible and A-1 = Adj A Det A    .

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