Question

If A is order 3 square matrix such that $|A| = 2$ then $|adj(adj(adjA))|$ is?
A. 512
B. 256
C. 64
D. None of these

Answer 1

Hint: In the given question, they have given us A matrix which is of order of 3 and $|A| = 2$ . To solve this question, we will be applying a theorem of the adjoint of a matrix i.e. given by
$ \Rightarrow |adj(A)| = |A{|^{n - 1}}$
where A is a square matrix and n is the order of that matrix. It is given that the order of A matrix is 3. Therefore $n = 3$

Complete step by step answer:

Let us see what is given to us? We have given a matrix whose order is 3 and $|A| = 2$.
$ \Rightarrow n = 3$
After that see what we have to find? We have to find the value of $|adj(adj(adjA))|$.
First of all, we will find the value of adjoint A. Applying the formula, we get
$ \Rightarrow |adj(A)| = |A{|^{n - 1}}$
Putting $n = 3$ in above equation we get,
 $ \Rightarrow |adj(A)| = |A{|^{3 - 1}}$
$ \Rightarrow |adj(A)| = |A{|^2}$
Putting the value of $|A| = 2$in above equation we get,
$ \Rightarrow |adj(A)| = {(2)^2}$
The square of 2 is 4 and by putting it in the above equation we get,
$ \Rightarrow |adj(A)| = 4$
We will again apply the formula on adjoint A i.e. adjoint of adjoint A is given by
$ \Rightarrow |adj(adjA)| = |adj(A){|^{n - 1}}$
Putting $n = 3$ in above equation we get,
 $ \Rightarrow |adj(adjA)| = |adj(A){|^{3 - 1}}$
$ \Rightarrow |adj(adjA)| = |adj(A){|^2}$
Putting the value of $|adj(A)| = 4$in above equation we get,
$ \Rightarrow |adj(adjA)| = {(4)^2}$
The square of 4 is 16 and by putting it in the above equation we get,
$ \Rightarrow |adj(adjA)| = 16$
We will again apply the formula on adjoint of adjoint A i.e. adjoint of adjoint A is given by
$ \Rightarrow |adj(adj(adjA)| = |adj(adjA){|^{n - 1}}$
Putting $n = 3$ in above equation we get,
 $ \Rightarrow |adj(adj(adjA)| = |adj(adjA){|^{3 - 1}}$
$ \Rightarrow |adj(adj(adjA)| = |adj(adjA){|^2}$
Putting the value of $|adj(adjA)| = 16$in above equation we get,
$ \Rightarrow |adj(adj(adjA)| = {(16)^2}$
The square of 16 is 256 and by putting it in the above equation we get,
$ \Rightarrow |adj(adj(adjA)| = 256$
The value of $|adj(adj(adjA))|$is 256.
So, the correct option is B.

Note: The common mistakes done by students are forgetting to subtract 1 from n, they directly use n i.e. the power n, not n-1 which is wrong, always remember to subtract 1 from n in the power.
Additional information: If A and b are the square matrices of the same order but both are non-singular matrix, then adjoint ab is given by
$ \Rightarrow adj(AB) = adjB \times adjA$.
 If A matrix is a square matrix and it is non-singular, then
$ \Rightarrow adj(adjA) = |A{|^{n - 2}}A$
If A is invertible i.e. its inverse exists, then
$ \Rightarrow ad{j^{}}{A^T} = {(adjA)^T}$