Q107 / 180 · COMEDK 2025 (25 May Shift) · Mathematics
If $A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]$ then $|\operatorname{adj} A|=$
- A$a^2$
- Correct answerB$a^6$
- C$a^9$
- D$a^{27}$
Solution
The matrix $A$ is a scalar matrix of order $n=3$, given by $A = aI$, where $I$ is the identity matrix of order $3$.
The determinant of $A$ is $|A| = \begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{vmatrix} = a^3$.
The property of the adjoint of a matrix states that $|\operatorname{adj} A| = |A|^{n-1}$, where $n$ is the order of the square matrix $A$.
Substituting $n=3$ and $|A| = a^3$ into the formula:
$|\operatorname{adj} A| = (a^3)^{3-1} = (a^3)^2 = a^6$.
Answer: $a^6$
The determinant of $A$ is $|A| = \begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{vmatrix} = a^3$.
The property of the adjoint of a matrix states that $|\operatorname{adj} A| = |A|^{n-1}$, where $n$ is the order of the square matrix $A$.
Substituting $n=3$ and $|A| = a^3$ into the formula:
$|\operatorname{adj} A| = (a^3)^{3-1} = (a^3)^2 = a^6$.
Answer: $a^6$
Hence, the correct answer is option: 'B'.