Matrices & Determinants
Determinant of Adjoint of Adjoint of Scaled Matrix
nta_pyq_2023_apr
Grade 12
Question:
If $A=\dfrac{1}{5!6!7!}\begin{pmatrix}5!&6!&7!\\6!&7!&8!\\7!&8!&9!\end{pmatrix}$, then $|\text{adj}(\text{adj}(2A))|$ is equal to
$2^{20}$
$2^8$
$2^{12}$
$2^{16}$
Step-by-Step Solution
Key Concept: Compute $|A|=2$. $|\text{adj}(\text{adj}(2A))|=|2A|^{(n-1)^2}=|2A|^4$.
$|2A|=16$. $|\text{adj}(\text{adj}(2A))|=16^4=2^{16}$.
Correct Answer: 4