Q75 / 180 · COMEDK 2022 · Mathematics
If $A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]$, then $|A|| \operatorname{adj} A \mid$ is equal to
- Correct answerA$a^9$
- B$a^{-9}$
- C$-a^9$
- D$2a^9$
Solution
$A$ is a $3 \times 3$ diagonal matrix $\Rightarrow |A| = a \cdot a \cdot a = a^3$
Using the property: $|A||\text{adj}\,A| = |A|^n$ for an $n \times n$ matrix:
$|A||\text{adj}\,A| = |A|^3 = (a^3)^3 = a^9$
Correct option: (1)
Using the property: $|A||\text{adj}\,A| = |A|^n$ for an $n \times n$ matrix:
$|A||\text{adj}\,A| = |A|^3 = (a^3)^3 = a^9$
Correct option: (1)
Hence, the correct answer is option: 'A'.