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**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

M_{11} = a _{22} , M_{12 }= a _{21}, M_{21} = a , , M_{22} = a_{11}

Also C_{11 }= a_{22} , C_{12} = -a_{21}, C_{21} = -a_{12} , C_{22} = a_{11}

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 = **

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) A^{2} = I implies A^{-1} = A

(x) If AB = C then (a) A = CB^{-1} (b) B = A^{-1}C, 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 :**

**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

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