Matrices & Determinants
Scaled Matrix — det(adj(adj P))
nta_pyq_2026_jan
Grade 12
Question:
Let $P=[p_{ij}]$ and $Q=[q_{ij}]$ be two square matrices of order 3 such that $q_{ij}=2^{(i+j-1)}p_{ij}$ and $\det(Q)=2^{10}$. Then the value of $\det(\text{adj}(\text{adj}\,P))$ is:
Step-by-Step Solution
Key Concept: Factor out row and column scalars: $\det(Q)=2^2\cdot2\cdot2^3\cdot\begin{vmatrix}p_{11}&p_{12}&p_{13}\\2p_{21}&2p_{22}&2p_{23}\\2^2p_{31}&2^2p_{32}&2^2p_{33}\end{vmatrix}=2^9\det(P)=2^{10}\Rightarrow\det(P)=2$.
$\det(P)=2$. $\det(\text{adj}(\text{adj}\,P))=16$.
Correct Answer: 2