Answer
Verified
396.3k+ views
Hint: Here we’ll use some properties of determinants and adjoint of square matrices like $\left| {{\text{AdjM}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{{\text{n - 1}}}}$and $\left| {{{\text{M}}^{\text{a}}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{\text{a}}}$, first we’ll find the value of |Adj${{\text{A}}^{\text{2}}}$| then again applying the same property will find the value of$\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|$to get the required answer.
Complete step by step answer:
Given data: A is a square matrix of order 3
As we all know that, if M is a square matrix of order n
Then, $\left| {{\text{AdjM}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{{\text{n - 1}}}}$
Similarly, we can also say that
$\left| {{\text{Adj(AdjM)}}} \right|{\text{ = (}}{\left| {\text{M}} \right|^{{\text{n - 1}}}}{{\text{)}}^{\text{2}}}$
Now, A is a matrix of order 3, so can conclude that
\[
\left| {{\text{Adj(AdjA)}}} \right|{\text{ = (}}{\left| {\text{A}} \right|^{{\text{3 - 1}}}}{{\text{)}}^{\text{2}}} \\
{\text{ = (}}{\left| {\text{A}} \right|^{\text{2}}}{{\text{)}}^{\text{2}}} \\
{\text{ = }}{\left| {\text{A}} \right|^{\text{4}}} \\
\]
Therefore it is applicable for \[{{\text{A}}^{\text{2}}}\] as it will also be a square matrix, concluding that
\[
\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = (}}{\left| {{{\text{A}}^{\text{2}}}} \right|^{{\text{3 - 1}}}}{{\text{)}}^{\text{2}}} \\
{\text{ = (}}{\left| {{{\text{A}}^{\text{2}}}} \right|^{\text{2}}}{{\text{)}}^{\text{2}}} \\
{\text{ = }}{\left| {{{\text{A}}^{\text{2}}}} \right|^{\text{4}}} \\
\]
Since we know that for a square matrix M of order n
$\left| {{{\text{M}}^{\text{a}}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{\text{a}}}$
\[
\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = }}{\left| {{{\text{A}}^{\text{2}}}} \right|^{\text{4}}} \\
{\left| {\text{A}} \right|^{\text{8}}} \\
\]
Therefore, option (C)${{\text{A}}^{\text{8}}}$ is the correct option
Note: An alternative solution for this question can be
Since we know that for a square matrix M of order n
$\left| {{{\text{M}}^{\text{a}}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{\text{a}}}$
Now, since A is also a square matrix
$\left| {{{\text{A}}^{\text{2}}}} \right|{\text{ = }}{\left| {\text{A}} \right|^{\text{2}}}$
Now, applying the same rule as the above solution
\[
\left| {{\text{(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = }}{\left( {{{\left| {\text{A}} \right|}^{\text{2}}}} \right)^{{\text{3 - 1}}}} \\
{\text{ = }}{\left| {\text{A}} \right|^{\text{4}}} \\
\]
Again using the same formula,
\[
\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = }}{\left( {{{\left| {\text{A}} \right|}^{\text{4}}}} \right)^{{\text{3 - 1}}}} \\
{\text{ = }}{\left| {\text{A}} \right|^{\text{8}}} \\
\]
Complete step by step answer:
Given data: A is a square matrix of order 3
As we all know that, if M is a square matrix of order n
Then, $\left| {{\text{AdjM}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{{\text{n - 1}}}}$
Similarly, we can also say that
$\left| {{\text{Adj(AdjM)}}} \right|{\text{ = (}}{\left| {\text{M}} \right|^{{\text{n - 1}}}}{{\text{)}}^{\text{2}}}$
Now, A is a matrix of order 3, so can conclude that
\[
\left| {{\text{Adj(AdjA)}}} \right|{\text{ = (}}{\left| {\text{A}} \right|^{{\text{3 - 1}}}}{{\text{)}}^{\text{2}}} \\
{\text{ = (}}{\left| {\text{A}} \right|^{\text{2}}}{{\text{)}}^{\text{2}}} \\
{\text{ = }}{\left| {\text{A}} \right|^{\text{4}}} \\
\]
Therefore it is applicable for \[{{\text{A}}^{\text{2}}}\] as it will also be a square matrix, concluding that
\[
\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = (}}{\left| {{{\text{A}}^{\text{2}}}} \right|^{{\text{3 - 1}}}}{{\text{)}}^{\text{2}}} \\
{\text{ = (}}{\left| {{{\text{A}}^{\text{2}}}} \right|^{\text{2}}}{{\text{)}}^{\text{2}}} \\
{\text{ = }}{\left| {{{\text{A}}^{\text{2}}}} \right|^{\text{4}}} \\
\]
Since we know that for a square matrix M of order n
$\left| {{{\text{M}}^{\text{a}}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{\text{a}}}$
\[
\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = }}{\left| {{{\text{A}}^{\text{2}}}} \right|^{\text{4}}} \\
{\left| {\text{A}} \right|^{\text{8}}} \\
\]
Therefore, option (C)${{\text{A}}^{\text{8}}}$ is the correct option
Note: An alternative solution for this question can be
Since we know that for a square matrix M of order n
$\left| {{{\text{M}}^{\text{a}}}} \right|{\text{ = }}{\left| {\text{M}} \right|^{\text{a}}}$
Now, since A is also a square matrix
$\left| {{{\text{A}}^{\text{2}}}} \right|{\text{ = }}{\left| {\text{A}} \right|^{\text{2}}}$
Now, applying the same rule as the above solution
\[
\left| {{\text{(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = }}{\left( {{{\left| {\text{A}} \right|}^{\text{2}}}} \right)^{{\text{3 - 1}}}} \\
{\text{ = }}{\left| {\text{A}} \right|^{\text{4}}} \\
\]
Again using the same formula,
\[
\left| {{\text{Adj(Adj}}{{\text{A}}^{\text{2}}}{\text{)}}} \right|{\text{ = }}{\left( {{{\left| {\text{A}} \right|}^{\text{4}}}} \right)^{{\text{3 - 1}}}} \\
{\text{ = }}{\left| {\text{A}} \right|^{\text{8}}} \\
\]
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE