Question

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

Answer 1

Hint: In this question it is given that A is a square matrix of order 3, then we have to find the value of $\left\vert Adj\left( AdjA^{2}\right) \right\vert$. So to find the solution we have to use one important formula, which is,
$\left\vert AdjA\right\vert =A^{\left( n-1\right) }$,…….(1)
where n is the order of the matrix.
So by using the above formula we will get our required solution.
Complete step-by-step solution:
Let, $AdjA^{2}=B$
Therefore, we can write,
$\left\vert Adj\left( AdjA^{2}\right) \right\vert =\left\vert AdjB\right\vert$.........(2)
Since, the order of the matrix A is $3\times3$, then the order of the matrix $A^{2}$ and B is also $3\times3$.
Therefore, from (2) we can write,
$\left\vert Adj\left( AdjA^{2}\right) \right\vert$
$=\left\vert AdjB\right\vert$
$=\left\vert B\right\vert^{\left( 3-1\right) }$ [ by using formula (1), and since, n=3]
$=\left\vert B\right\vert^{2}$
$=\left\vert AdjA^{2}\right\vert^{2}$
$=\left( \left\vert A^{2}\right\vert^{3-1} \right)^{2}$
$=\left( \left\vert A^{2}\right\vert^{2} \right)^{2}$
$=\left\vert A^{2}\right\vert^{2\times 2}$
$=\left\vert A^{2}\right\vert^{4}$
$=\left\vert A\right\vert^{2\times 4}$
$=\left\vert A\right\vert^{8}$
Hence, the correct option is option C.
Note: In the solution part we take the order of the matrix $A^{2}$ and $AdjA^{2}$ as $3\times3$, so for this you have to remember that when you perform any operations( e.g- addition, subtraction, multiplication) in two square matrix of same order, then the order of the resultant matrix is same as the multiplied matrices.
Also if the order of a matrix $n\times n$ then the matrix is called a square matrix of order n.