Q169 / 270 · NDA 2026 (Phase 1) · Mathematics
If $M=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$, then what is the value of $|M|\,|\text{adj}M|$ ?
- A$8$
- B$64$
- C$256$
- Correct answerD$512$
Solution
Given $M = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$
The determinant of the diagonal matrix $M$ is $|M| = 2 \times 2 \times 2 = 8$.
For a square matrix of order $n$, the determinant of its adjoint is given by $|\text{adj}M| = |M|^{n-1}$.
Since the order of $M$ is $n = 3$, we have $|\text{adj}M| = |M|^{3-1} = |M|^2$.
The required value is $|M| \, |\text{adj}M| = |M| \times |M|^2 = |M|^3$.
Substituting $|M| = 8$, we get $8^3 = 512$.
Answer: $512$
The determinant of the diagonal matrix $M$ is $|M| = 2 \times 2 \times 2 = 8$.
For a square matrix of order $n$, the determinant of its adjoint is given by $|\text{adj}M| = |M|^{n-1}$.
Since the order of $M$ is $n = 3$, we have $|\text{adj}M| = |M|^{3-1} = |M|^2$.
The required value is $|M| \, |\text{adj}M| = |M| \times |M|^2 = |M|^3$.
Substituting $|M| = 8$, we get $8^3 = 512$.
Answer: $512$
Hence, the correct answer is option: 'D'.