Question

For a matrix A of order $3\times 3$, $X=\underbrace{\left| adj\left( adj\left( adj.....\left( A \right) \right) \right) \right|}_{\text{r times}}$. Find X.

Answer 1

Hint: To solve this question, we should know the relation between the determinants of the matrix and its adjoint matrix. We know that for a square matrix A of order n, the relation between the determinants is $\left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}}$. Using this relation, we can expand the above-given determinant r times to get the value of X.

Complete step by step answer:
We are given a matrix A of order $3\times 3$. We are asked to find the value of X which is defined by the determinant of multiple adjoints of a matrix A.
We know that for a square matrix A of order n, the relation between the determinants is $\left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}}$.
In our question, the value of n is 3.
So, we can write that the order of matrices $adj\left( A \right),adj\left( adj\left( A \right) \right).....$is also 3. We can write that
$\left| adj\left( A \right) \right|={{\left| A \right|}^{3-1}}={{\left| A \right|}^{2}}$
 Let us consider $\left| adj\left( adj\left( A \right) \right) \right|$
We can write that
$\left| adj\left( adj\left( A \right) \right) \right|={{\left| adj\left( A \right) \right|}^{2}}={{\left( {{\left( \left| A \right| \right)}^{2}} \right)}^{2}}={{\left| A \right|}^{4}}$
Similarly, let us consider $\left| adj\left( adj\left( adj\left( A \right) \right) \right) \right|={{\left| adj\left( adj\left( A \right) \right) \right|}^{2}}={{\left( {{\left| A \right|}^{4}} \right)}^{2}}={{\left| A \right|}^{8}}$
If we observe the above equations, we can write that for every increase of the adjoint, the determinant value is increasing by a power of 2. We can establish a relation that
$\underbrace{\left| adj\left( adj\left( adj.....\left( A \right) \right) \right) \right|}_{\text{r times}}={{\left( {{\left| A \right|}^{2}} \right)}^{r}}={{\left| A \right|}^{2r}}$
We know that the above obtained value is X.
So, we can write that
$X={{\left| A \right|}^{2r}}$
$\therefore $The required value X is $X={{\left| A \right|}^{2r}}$.

Note:
Students can make a mistake while writing the relation by generalizing the value of r. If this is a multiple-choice question, there will be options like $X={{\left| A \right|}^{2r-2}}$, $X={{\left| A \right|}^{2r+2}}$ to misguide the students. It is always a good practice to verify the result that we get in these type of questions. For example, when $r=1$ we get the value of X as $x=\left| adj\left( A \right) \right|={{\left| A \right|}^{2}}$ which also verifies the result that $X={{\left| A \right|}^{2r}}$ is correct. Likewise, we should verify the general result that we get in these types of questions.