Let 0 and A=[[|gathered 3 - 2 gathered]]|.

Question

Let $\alpha \beta 
e 0$ and $A=[[|\begin{gathered} \beta  \quad  \alpha  \quad  3 \ \alpha  \quad  \alpha  \quad  \beta \ -\beta  \quad  \alpha  \quad  2\alpha \end{gathered}]]|$. If $B=[[|\begin{gathered} 3\alpha  \quad  -9 \quad  3\alpha \ -\alpha  \quad  7 \quad  -2\alpha \ -2\alpha  \quad  5 \quad  -2\beta \end{gathered}]]|$ is the matrix of cofactors of the elements of $A$, then $\det (AB)$ is equal to :

Accepted Answer

$A=[[|\begin{gathered} \beta  \quad  \alpha  \quad  3 \ \alpha  \quad  \alpha  \quad  \beta \ -\beta  \quad  \alpha  \quad  2\alpha \end{gathered}]]|,B=[[|\begin{gathered} 3\alpha  \quad  -9 \quad  3\alpha \ -\alpha  \quad  7 \quad  -2\alpha \ -2\alpha  \quad  5 \quad  -2\beta \end{gathered}]]|$

Cofactor of $A$-matrix is

$=[[|\begin{gathered} 2{\alpha }^{2}-\alpha \beta  \quad  -(2{\alpha }^{2}+{\beta }^{2}) \quad  {\alpha }^{2}+\alpha \beta \ -(2{\alpha }^{2}-3\alpha ) \quad  (2\alpha \beta +3\beta ) \quad  -(2\alpha \beta ) \ \alpha \beta -3\alpha  \quad  -({\beta }^{2}-3\alpha ) \quad  \beta \alpha -{\alpha }^{2} \end{gathered}]]|$

which is equal to matrix $B$

So, by comparing elements of two matrix

$\begin{gathered} \Rightarrow \alpha \beta -3\alpha =-2\alpha \ \Rightarrow \alpha \beta -\alpha =0 \ \Rightarrow \alpha (\beta -1)=0 \ \Rightarrow \alpha =0\beta =1[\because \alpha 0] \ \Rightarrow \beta =1 \ -{\beta }^{2}+3\alpha =5 \ \Rightarrow 3\alpha =6 \ \Rightarrow \alpha =2 \ A=[[|\begin{array}{ccc}1 \quad  2 \quad  3 \ 2 \quad  2 \quad  1 \ -1 \quad  2 \quad  4\end{array}]]| \ Det(AB)=|A||B|=|A||(adjA{)}^{}| \ =|A|\cdot |A{|}^{2} \ =|A{|}^{3} \ =(6-18+18{)}^{3} \ ={6}^{3} \ =216 \end{gathered}$

Solutions · 1