Matrices & Determinants
Inverse of Matrix Product
nta_pyq_2025_apr
Grade 12
Question:
If $A$, $B$, and $\left(\text{adj}(A^{-1}) + \text{adj}(B^{-1})\right)$ are non-singular matrices of same order, then the inverse of $A\left(\text{adj}(A^{-1}) + \text{adj}(B^{-1})\right)^{-1} B$ is equal to:
$AB^{-1} + A^{-1}B$
$\text{adj}(B^{-1}) + \text{adj}(A^{-1})$
$\dfrac{AB^{-1}}{|A|} + \dfrac{BA^{-1}}{|B|}$
$\dfrac{1}{|AB|}(\text{adj}(B) + \text{adj}(A))$
Step-by-Step Solution
Key Concept: Use $\text{adj}(A^{-1}) = \frac{A}{|A|}$ (since adj(M) = |M|M^{-1}). Simplify the inner sum, then find the inverse of the full product.
$(A(\text{adj}(A^{-1}) + \text{adj}(B^{-1}))^{-1}B)^{-1} = B^{-1}(\text{adj}(A^{-1}) + \text{adj}(B^{-1}))A^{-1} = \frac{B^{-1}}{|A|} + \frac{A^{-1}}{|B|} = \frac{1}{|AB|}(\text{adj}(B) + \text{adj}(A))$.
Correct Answer: $\dfrac{1}{|AB|}(\text{adj}(B) + \text{adj}(A))$