Matrices & Determinants
Matrices and Determinants
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Grade None
Question:
If $A$ & $B$ are two invertible matrices of same order, then $\text{Adj}(AB) =$
$|A||B|A^{-1}B^{-1}$
$\text{adj}(A) \cdot \text{adj}(B)$
$|A||B|(AB)^{-1}$
$\text{adj}(B) \cdot \text{adj}(A)$
Step-by-Step Solution
Key Concept: The adjugate satisfies $\text{Adj}(M) = |M|M^{-1}$, and for products, $\text{Adj}(AB) = \text{Adj}(B)\text{Adj}(A)$ (order reverses).
We use the fundamental property that $\text{Adj}(M) = |M|M^{-1}$ for any invertible matrix $M$. For the product $AB$, we have $\text{Adj}(AB) = |AB|(AB)^{-1}$. Since $|AB| = |A||B|$ and $(AB)^{-1} = B^{-1}A^{-1}$, we get $\text{Adj}(AB) = |A||B|B^{-1}A^{-1}$. This can also be written as $|A||B|(AB)^{-1}$ (option 3). Additionally, using $\text{Adj}(AB) = \text{Adj}(B)\text{Adj}(A)$ (the adjugate reverses order like inverse), option 4 is also correct. Both options 3 and 4 are equivalent and correct.
Correct Answer: 3,4