Question

What is adjoint and inverse of a matrix?

Answer 1

Adjoint and inverse of a matrix.

• The adjoint of a square matrix $A={\left[a_{ij}\right]}_{n\times n}$ is defined as the transpose of the matrix ${\left[A_{ij}\right]}_{nxn}$, where $A_{ij}$ is the cofactor of the element $a_{ij}$. Adjoint of the matrix $A$ is denoted by $adjA$.

To find the inverse of a matrix $A$, i.e. $A^{-1}$ , first define the adjoint of a matrix.

Let $A$ be an $n\times n$ matrix. The $(i,j)$ cofactor of $A$ is defined to be

$A_{ij}={(-1)}^{ij}det\left(M_{ij}\right)$

where

• $M_{ij}$ is the ${(i,j)}^{th}$ minor matrix obtained from $A$ after removing the $i^{th}$ row and $j^{th}$ column.

Let’s consider the n x n matrix $A=\left(A_{ij}\right)$ and define the $n\times n$ matrix $Adj\left(A\right)=A^T$ .

Hence, the matrix $Adj\left(A\right)$ is called the adjoint of matrix $A$The inverse is defined only for non-singular square matrices and is denoted by $A^{-1}$.