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# Adjoint of a Square Matrix

We recall the properties of the cofactors of the elements of a square matrix.

We recall the properties of the cofactors of the elements of a square matrix. Let A be a square matrix of by order n whose determinant is denoted  | A | or det ( A). Let aij be the element sitting at the intersection of the ith row and jth column of A. Deleting the ith row and  jth column of A, we obtain  a sub-matrix of order (n Ôłĺ1). The determinant of this sub-matrix is called minor of the element aij . It is denoted by Mij .The product of Mij and (Ôłĺ1)i+ j is called cofactor of the element aij . It is denoted by Aij. Thus the cofactor of aij is Aij = (Ôłĺ1)i+j Mij.

An important property connecting the elements of a square matrix and their cofactors is that the sum of the products of the entries (elements) of a row and the corresponding cofactors of the elements of the same row is equal to the determinant of the matrix; and the sum of the products of the entries (elements) of a row and the corresponding cofactors of the elements of any other row is equal to 0. That is, where |A| denotes the determinant of the square matrix A. Here |A| is read as ÔÇťdeterminant of A ÔÇŁ and not as ÔÇť modulus of A ÔÇŁ. Note that | A| is just a real number and it can also be negative. For instance, we have = 2(1Ôłĺ 2) Ôłĺ1(1Ôłĺ 2) +1(2 Ôłĺ 2) = Ôłĺ2 +1+ 0 = Ôłĺ1.

### Definition 1.1

Let A be a square matrix of order n. Then the matrix of cofactors of A is defined as the matrix obtained by replacing each element aij of A with the corresponding cofactor Aij. The adjoint matrix of A is defined as the transpose of the matrix of cofactors of A. It is denoted by adj A.

Note

adj A is a square matrix of order n  and  adj A = [ Aij ]T = [(Ôłĺ1)i + j Mij ]T

In particular, adj A of a square matrix of order 3 is given below: ### Theorem 1.1

For every square matrix A of order n , A(adj A) = (adj A) A = |A| In .

### Proof

For simplicity, we prove the theorem for n = 3 only. By using the above equations, we get where I3 is the identity matrix of order 3.

So, by equations (1) and (2), we get

### Note

If A is a singular matrix of order n , then | A | = 0 and so A(adj A) = (adj A) A = On , where On denotes zero matrix of order n.

### Example 1.1

If A = , verify that A (adj A) =  (adj A) A = | A | I3 .

Solution Tags : Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants
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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Adjoint of a Square Matrix | Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix