Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

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INVERSE OF A MATRIX
Minors and Cofactors of the elements of a determinant.
The minor of an element aij of a determinant A is denoted by Mij and is the determinant obtained from A by deleting the row and the column where aij occurs.
The cofactor of an element aij with minor Mij is denoted by Cij and is defined as

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Thus, cofactors are signed minors.

In the case of  Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev , we have

M11 = a 22 , M12 = a 21, M21 = a , , M22 = a11
Also C11 = a22 , C12 = -a21, C21 = -a12 , C22 = a11

In the case of Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev we have

   Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
   Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
 Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Adjoint of a square matrix.
The transpose of the matrix got by replacing all the elements of a square matrix A by their corresponding cofactors in | A | is called the Adjoint of A or Adjugate of A and is denoted by Adj A.

Thus, AdjA =  Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

 

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
 

(ii) Adj I = I, where I is the unit matrix.
(iii) A(AdjA) = (Adj A) A = | A | I
(iv) Adj (AB) = (Adj B) (Adj A)
(v) If A is a square matrix of order 2, then |AdjA| = |A|
     If A is a square matrix of order 3, then |Adj A| = |A|2

Example 1 Write the Adjoint of the matrix A =  Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Solution : 
Adj A = Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

 

Example 2 
 Find the Adjoint of the matrix A =  
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Solution :
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Now,

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Hence
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Inverse of a non singular matrix.
The inverse of a non singular matrix A is the matrix B such that AB = BA = I. B is then called the inverse of A and denoted by A-1 .
Note
(i) A non square matrix has no inverse.
(ii) The inverse of a square matrix A exists only when |A| ≠ 0 that is, if A is a singular matrix then A-1 does not exist.
(iii) If B is the inverse of A then A is the inverse of B. That is B = A-1 ⇒ A = B-1.
(iv) A A-1 = I = A-1 A
(v) The inverse of a matrix, if it exists, is unique. That is, no matrix can have more than one inverse.
(vi) The order of the matrix A-1 will be the same as that of A.
(vii) I-1 = I
(viii) (AB)-1 = B-1 A-1 , provided the inverses exist.
(ix) A2 = I implies A-1 = A
(x) If AB = C then (a) A = CB-1 (b) B = A-1C, provided the inverses exist.
(xi) We have seen that

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

This suggests that

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

(xii) (A-1 ) -1 = A, provided the inverse exists.

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Example 3​
Find the inverse of Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev if it exists


Solution :​  

  Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Example 4
Show that the inverses of the following do not exist :​


Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Solution :

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev
 Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev


Example 5
Find the inverse of Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev if it exists.

Solution :​

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

We have, Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Now, the cofactors are
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Hence
Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Example 6

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Inverse of a Matrix - Matrices and Determinants, Business Mathematics & Statistics B Com Notes | EduRev

Since A and B are square matrices and AB = I, A and B are inverse of each other

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