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.