Adjoint and Inverse of a Matrix - Practice Questions & MCQ

Adjoint of a Matrix

Adjoint of a matrix A is the transpose of cofactor matrix of the matrix A . Cofactor matrix of matrix A is a matrix which has same order as that of A and has element $C_{ij}$ in place of $a_{ij}$. E.g.,

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

and let cofactor of every element is

$\left[\begin{array}{lll}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$

then Adjoint of A is

$A^{\prime }={\left[\begin{array}{lll}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]}^{\prime }=\left[\begin{array}{lll}{C}_{11}& C_{21}& C_{31}\\ C_{12}& C_{22}& C_{32}\\ C_{13}& C_{23}& C_{33}\end{array}\right]$
 

Properties of adjoint of a matrix

1. If A is a square matrix of order n, then
$(\mathrm{Adj}A)A=A(\mathrm{Adj}A)=|A|{\mathbb{I}}_n$, or product of a matrix and its adjoint is commutative.
Proof:
Let, $\mathrm{A}=\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then adj $\mathrm{A}=\left[\begin{array}{lll}{A}_{11}& A_{21}& A_{31}\\ A_{12}& A_{22}& A_{32}\\ A_{13}& A_{23}& A_{33}\end{array}\right]$
where, $A_{ij}$ is co - factor of $a_{ij}$
Since the sum of the product of elements of a row (or a column) with corresponding cofactors is equal to $|\mathrm{A}|$ and otherwise zero, we have

$\mathrm{A}(\mathrm{adj}\mathrm{A})=\left[\begin{array}{ccc}|A|& 0& 0\\ 0& |A|& 0\\ 0& 0& |A|\end{array}\right]=|\mathrm{A}|\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=|\mathrm{A}|\mathrm{I}$
If $A$ is a singular matrix of order $n$, then

$(\mathrm{Adj}A)A=A(\mathrm{Adj}A)=O\text{ (null matrix })(\text{ As }|A|=0)$

2. If $A$ be non-singular square matrix of order $n$, then $|\mathrm{Adj}A|=|A{|}^{n-1}$

Proof:

$\mathrm{A}(\mathrm{Adj}\mathrm{A})=|\mathrm{A}|{\mathrm{I}}_{\mathrm{n}}$
Taking determinants on both sides

$\begin{array}{rl}& |\mathrm{A}(\mathrm{Adj}\mathrm{A})|=\left||\mathrm{A}|{\mathrm{I}}_{\mathrm{n}}\right|\\ & |\mathrm{A}||(\mathrm{Adj}\mathrm{A})|=|\mathrm{A}{|}^{\mathrm{n}}\\ & |(\mathrm{adj}\mathrm{A})|=|\mathrm{A}{|}^{\mathrm{n}-1}\end{array}$

3. If $A$ and $B$ are square matrices of order $n$, then, $\mathrm{adj}(AB)=(\mathrm{adj}B)(\mathrm{adj}A)$
4. If $A$ is a square matrix of order $n$, then, $(\mathrm{adj}A{)}^{\prime }=\mathrm{adj}{A}^{\prime }$

Properties of adjoint of matrix

4. If $A$ be a square non-singular matrix of order $n$, then $\mathrm{adj}(\mathrm{adj}A)=|A{|}^{n-2}A$

Proof:

$\begin{array}{rl}& A(\mathrm{adj}A)=|A|{\mathbb{I}}_n\\ & \mathrm{replace}A\mathrm{by}\mathrm{adj}A,\text{ then }\\ & (\mathrm{adj}A)(\mathrm{adj}(\mathrm{adj}A))=|\mathrm{adj}A|{\mathbb{I}}_n=|A{|}^{\mathrm{n}-1}{\mathbb{I}}_{\mathrm{n}}\\ & \text{ Pre }-\mathrm{multiplying}\mathrm{both}\mathrm{sides}\text{ by matrix }A\text{, then }\\ & A(\mathrm{adj}A)(\mathrm{adj}(\mathrm{adj}A))=A{\mathbb{I}}_n|A{|}^{\mathrm{n}-1}=A|A{|}^{\mathrm{n}-1}\\ & |A|{\mathbb{I}}_{\mathrm{n}}(\mathrm{adj}(\mathrm{adj}A))=A|A{|}^{\mathrm{n}-1}\\ & (\mathrm{adj}(\mathrm{adj}A))=A|A{|}^{\mathrm{n}-2}=|A{|}^{\mathrm{n}-2}\mathrm{A}\end{array}$

5. If A is non-singular square matrix, then, $|\mathrm{adj}(\mathrm{adj}\mathrm{A})|=|\mathrm{A}{|}^{(\mathrm{n}-1{)}^2}$

Proof: from the previous property, we know that

$\mathrm{adj}(\mathrm{adj}\mathrm{A})=|\mathrm{A}{|}^{(\mathrm{n}-2)}\mathrm{A}$
Taking determinant on both sides,

$\begin{array}{rl}& |\mathrm{adj}(\mathrm{adj}\mathrm{A})|=\left||\mathrm{A}{|}^{(\mathrm{n}-2)}\mathrm{A}\right|=|\mathrm{A}{|}^{\mathrm{n}(\mathrm{n}-2)}|\mathrm{A}|=|\mathrm{A}{|}^{(\mathrm{n}-1{)}^2}\\ & \text{ (using }|kA|=k^n|A|)\end{array}$

6. If $A$ be a square matrix of order $n$ and $m$ be any natural number, then $\left(\mathrm{adj}{A}^m\right)=(\mathrm{adj}A{)}^m$
7. If $A$ is a square matrix of order $n$ and $k$ be a scalar, then, $\mathrm{adj}(kA)=k^{n-1}\cdot (\mathrm{adj}A)$