Adjoint of a Square Matrix

Adjoint 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 a_ij be the element sitting at the intersection of the i^th row and j^th column of A. Deleting the i^th row and  j^th column of A, we obtain  a sub-matrix of order (n −1). The determinant of this sub-matrix is called minor of the element a_ij . It is denoted by M_ij .The product of M_ij and (−1)^{i+ j} is called cofactor of the element a_ij . It is denoted by A_ij. Thus the cofactor of a_ij is A_ij = (−1)^i+j M_ij.

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 “determinant of A ” and not as “ modulus of A ”. Note that | A| is just a real number and it can also be negative. For instance, we have 

 = 2(1− 2) −1(1− 2) +1(2 − 2) = −2 +1+ 0 = −1.

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 a_ij of A with the corresponding cofactor A_ij. 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 = [ A_ij ]^T = [(−1)^{i + j} M_ij ]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| I_n .

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

A(adj A) = (adj A) A = |A| I₃.

Note

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

Example 1.1

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

Solution