Question

Let A be a 2x2 matrix
Statement- 1: adj ( adj A ) = A
Statement-2: | adj A | = | A |

Select the correct choice:
${\text{A}}{\text{. Statement 1 is false, Statement 2 is true; Statement 2 is correct explanation for Statement 1}} \\\ {\text{B}}{\text{. Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1}} \\\ {\text{C}}{\text{. Statement 1 is true, Statement 2 is false}} \\\ {\text{D}}{\text{. Statement 1 is false, Statement 2 is true}} \\\$

Answer 1

Hint- In the question, we will use the formula $adj\left( {adjA} \right) = {\left| A \right|^{n - 2}}A$ to prove the Statement-1 and then use $\left| {adjA} \right| = {\left| A \right|^{n - 1}}$ to solve the Statement-2. This helps us simplify the question and reach the answer easily.

Complete step-by-step answer:
We know that $adj\left( {adjA} \right) = {\left| A \right|^{n - 2}}A$

And we also are given a 2x2 matrix, so n = 2

$\Rightarrow adj\left( {adjA} \right) = {\left| A \right|^{2 - 2}}A = {\left| A \right|^0}A$
$\Rightarrow adj\left( {adjA} \right) = A$

So, Statement-1 is correct

Now, we also know that $\left| {adjA} \right| = {\left| A \right|^{n - 1}}$
And n=2

$\Rightarrow \left| {adjA} \right| = {\left| A \right|^{2 - 1}} = \left| A \right|$

So, the Statement-2 is also correct

But Statement-2 does not explain Statement-2. So, option B is correct.

Note- Whenever we face such types of problems the main point to remember is that we need to have a good grasp over Matrices and Determinants. In this particular question we need to directly use the formulas discussed above in order to solve the question as otherwise if we try to prove it from scratch, then it will take a lot of time and is very prone to error.