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

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



Thus, cofactors are signed minors.

In the case of  , 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 we have

   
   

 

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 = 

 

(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 = 

Solution : 
Adj A = 

Example 2 
 Find the Adjoint of the matrix A =  

Solution :


Now,

Hence


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





This suggests that



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




Example 3​
Find the inverse of if it exists


Solution :​  

 

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



Solution :


 

Example 5
Find the inverse of if it exists.

Solution :​



We have,

Now, the cofactors are


Hence


Example 6







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