Matrices & Determinants
Matrix of Cofactors — det(AB)
nta_pyq_2024_apr
Grade 12
Question:
Let $\alpha\beta\neq0$ and $A=\begin{bmatrix}\beta&\alpha&3\\\alpha&\alpha&\beta\\-\beta&\alpha&2\alpha\end{bmatrix}$. If $B=\begin{bmatrix}3\alpha&-9&3\alpha\\-\alpha&7&-2\alpha\\-2\alpha&5&-2\beta\end{bmatrix}$ is the matrix of cofactors of the elements of $A$, then $\det(AB)$ is equal to:
Step-by-Step Solution
Key Concept: If $B$ is the cofactor matrix of $A$, then $AB=|A|\cdot I$ (for square matrices), so $\det(AB)=|A|^3$. Alternatively, $\det(AB)=\det(A)\cdot\det(B)=\det(A)\cdot|A|^{n-1}=|A|^n$ for $n=3$.
$\alpha=2$, $\beta=1$, $|A|=6$. $\det(AB)=216$.
Correct Answer: 2