Matrices & Determinants
Adjoint inverse identity
MJMT_Full_Test_04
Grade 12
Question:
Let $P=\begin{bmatrix}0&2&\lambda\\2&3&1\\1&\mu&3\end{bmatrix}$ and $\text{Adj}(P)=\begin{bmatrix}10&-7&-1\\-5&-1&2\\-5&2&-4\end{bmatrix}$. Then $\left|(\text{adj}P)^{-1}+14\,\text{adj}(P^{-1})\right|$ equals
Step-by-Step Solution
Key Concept: $(\text{adj}P)^{-1}=P/|P|$ and $\text{adj}(P^{-1})=P/|P|$. So $(\text{adj}P)^{-1}+14\text{adj}(P^{-1})=15P/|P|$.
$|(\text{adj}P)^{-1}+14\,\text{adj}(P^{-1})|=15$.
Correct Answer: 1