We recall the properties of the cofactors of the elements 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

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.

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

**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:

For every square
matrix *A *of order *n *, *A*(adj *A*) = (adj *A*) *A *= |*A| I _{n} *.

For simplicity, we prove the theorem for *n *= 3 only.

By using the above equations, we get

where *I*_{3} is the identity matrix of order 3.

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

*A*(adj *A*) = (adj *A*) *A *= |*A| I*_{3}.

If *A *is a singular matrix of order *n *, then | A | = 0 and so *A*(adj
*A*) = (adj *A*) *A *= *O _{n} *,
where

If A = , verify that A (adj A) = (adj A) A = | A | I_{3} .

**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

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