The inverse of a square matrix obtaining by using elementary row or column operations.

Adjoint of a matrix

Consider A = $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$

Let A_ij = Cofactor of a_ij   $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$  $\leftrightarrow$  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{12}}& {\text{A}}_{\text{13}}\\ {\text{A}}_{\text{21}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{23}}\\ {\text{A}}_{\text{31}}& {\text{A}}_{\text{32}}& {\text{A}}_{\text{33}}\end{array}\right]$   

Transpose of matrix  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{12}}& {\text{A}}_{\text{13}}\\ {\text{A}}_{\text{21}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{23}}\\ {\text{A}}_{\text{31}}& {\text{A}}_{\text{32}}& {\text{A}}_{\text{33}}\end{array}\right]$  =  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}& {\text{A}}_{\text{31}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{32}}\\ {\text{A}}_{\text{13}}& {\text{A}}_{\text{23}}& {\text{A}}_{\text{33}}\end{array}\right]$ 

According to the definition, adj A =  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}& {\text{A}}_{\text{31}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{32}}\\ {\text{A}}_{\text{13}}& {\text{A}}_{\text{23}}& {\text{A}}_{\text{33}}\end{array}\right]$ 

Adjoint of a matrix

Let A = $\left[\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right]$

The cofactors of the elements of A are

A₁₁ = d, A₁₂ = -c, A₂₁ = -b, A₂₂ = a

A = $\left[\begin{array}{cc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}\end{array}\right]$ = $\left[\begin{array}{cc}\text{d}& \text{–b} \\ \text{–c}& \text{a}\end{array}\right]$

Theorem 1

If A is any given square matrix of order n, then:

A(adj A) = (adj A) A = |A| I, where I is the identity matrix of order n.

Singular matrix:

A square matrix is said to be singular if its determinant is zero.

Non-singular matrix:

A square matrix is said to be non-singular if its determinant is not zero.

Theorem 2

If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.

Claim: |AB| = |A||B|

Theorem 3

The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B| , where A and B are square matrices of the same order.

Theorem 4

If A is a square matrix of order n, then |(Adj A)| = |A|^n-1.

Inverse of a square matrix:

A square matrix, A, is said to be an invertible matrix if there exists a square matrix, B, such that AB = BA = I.

If a square matrix, A, is invertible, then it is a non-singular matrix.

Theorem 5

If A is a non-singular matrix, then A is invertible and A^-1 = $\frac{\text{Adj A}}{\text{Det A}}$   .