explain the question---**Exam Information:**
JEE Main 2024 (Online) 4th April Morning Shift
Question Type: MCQ (Single Correct Answer)
Marking: +4 for correct, -1 for incorrect
**Question Stem:**
Let $\alpha \in (0, \infty)$ and $A = \begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. If $\det (\operatorname{adj} (2A - A^T) \cdot \operatorname{adj} (A - 2A^T)) = 2^8$, then $(\det(A))^2$ is equal to:
**Options:**
A. 16
B. 36
C. 49
D. 1
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We have a JEE Main 2024 matrix problem. Given a 3 by 3 matrix A with parameter alpha, and a condition involving determinants of adjoints. We need to find the square of determinant of A. The key formula we'll use is that for an n by n matrix M, the determinant of its adjoint equals determinant of M raised to power n minus 1.
Let's apply the determinant properties. For any matrices M and N, determinant of M times N equals determinant of M times determinant of N. For a 3 by 3 matrix M, determinant of adjoint of M equals determinant of M squared. Using these properties, we transform the given condition step by step to get the final form.
Now let's calculate the required matrix operations. First, we find the transpose of matrix A. Then we compute 2A minus A transpose, which gives us a matrix with elements involving alpha. Next, we calculate A minus 2A transpose, which also contains alpha terms. These two matrices will be used to find the determinants in the next step.
Now we calculate the determinants. For the first matrix, expanding along the second row gives us negative one plus three alpha. For the second matrix, expanding along the second column gives us one plus three alpha. Substituting into our equation, we get one plus three alpha to the fourth power equals two to the eighth power, which equals four to the fourth power. Therefore, one plus three alpha equals four, so alpha equals one.
Finally, with alpha equals one, we substitute into matrix A and calculate its determinant. Expanding along the first row, we get determinant of A equals negative four. Therefore, determinant of A squared equals negative four squared, which equals sixteen. The answer is option A, sixteen.