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
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 =
Adj A =
Find the Adjoint of the matrix A =
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 .
(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.
Find the inverse of if it exists
Show that the inverses of the following do not exist :
Find the inverse of if it exists.
Now, the cofactors are
Since A and B are square matrices and AB = I, A and B are inverse of each other