Question

If $A$ is a square matrix of order $3$, then $\left| {Adj(Adj{A^2})} \right| =$
A. ${\left| A \right|^2}$
B. ${\left| A \right|^4}$
C. ${\left| A \right|^8}$
D. ${\left| A \right|^{16}}$

Answer 1

We have given that $A$ is matrix of order $3$ and we have to find value of $\left| {Adj(Adj{A^2})} \right|$. Firstly we have to find the result which gives the relation between $\left| {AdjA} \right|$ and $\left| A \right|$. This result helps us to find value of $\left| {AdjA} \right|$. Then we find relation between $\left| {Adj(AdjA)} \right|$ and $\left| A \right|$. This result help us to find value of $\left| {Adj(AdjA)} \right|$.
At the end we will be able to find the value of $\left| {Adj(Adj{A^2})} \right|$.

Complete step by step answer:
We have given that $A$ is a square matrix. This means that the number of rows in the matrix is equal to the number of columns of the matrix.
Order of the matrix is $3$ so the number of rows and number of columns is equal to $3$.
We have to find the value of $\left| {Adj(Adj{A^2})} \right|$.
We know that $\left| {AdjA} \right| = {\left| A \right|^{n - 1}}$ where $n$ is the order of the matrix.
$\left| {Adj(AdjA)} \right| = {\left| A \right|^{{{(n - 1)}^2}}}$
$\left| {Adj(Adj{A^2})} \right| = {\left| {{A^2}} \right|^{{{(3 - 1)}^2}}}$
$= {\left| {{A^2}} \right|^{{2^2}}} = {\left| {{A^2}} \right|^4}$
Now $\left| {{A^2}} \right| = {\left| A \right|^2}$
Therefore ${\left| {{A^2}} \right|^4} = {\left| A \right|^8}$
So $\left| {Adj(Adj{A^2})} \right| = {\left| A \right|^8}$
Hence the result is $\left| {Adj(Adj{A^2})} \right| = {\left| A \right|^8}$

Option (C) is correct.

Note: Determinant of Matrix : Every square matrix is associated by a real number which is called determinant of that matrix.
Adjoint of matrix : Adjoint of matrix is transpose of cofactor of that matrix.
Order of the matrix: The number of rows and number of columns of the matrix is called order of the matrix. Rows are listed as first, second and third and column also listed as first, second and third.