Let B=[[|gathered 1 3 1 2 3 4 gathered]]|, 2 be the adjoint of a matrix A and |A|=2.

Question

Let $B=[[|\begin{gathered} 1 \quad  3 \quad  \alpha \ 1 \quad  2 \quad  3 \ \alpha  \quad  \alpha  \quad  4 \end{gathered}]]|,\alpha >2$ be the adjoint of a matrix $A$ and $|A|=2$. Then 
$[\begin{gathered} \alpha  \quad  -2\alpha  \quad  \alpha \end{gathered}]B[[|\begin{gathered} \alpha \ -2\alpha \ \alpha \end{gathered}]]|$ is equal to :

Accepted Answer

$$
B=\left[\[\begin{array}{lll} 1 & 3 & \alpha \ 1 & 2 & 3 \ \alpha & \alpha & 4 \end{array}$$\right], \alpha>2
\]

And $\operatorname{adj}(A)=B,|A|=2$

$$
\begin{aligned}
& \Rightarrow|\operatorname{adj}(A)|=|B| \\
& \Rightarrow 2^2=(8-3 \alpha)-3(4-3 \alpha)+\alpha(-\alpha) \\
& \Rightarrow \alpha^2-6 \alpha+8=0
\end{aligned}
$$

$$
\begin{aligned}
\Rightarrow & (\alpha-4)(\alpha-2)=0 \\
& \alpha=4,2 \text { but } \alpha>2 \text { so } \alpha=4
\end{aligned}
$$

Now

$$
\begin{aligned}
& {\left[\[\begin{array}{ccc} \alpha & -2 \alpha & \alpha \end{array}\end{aligned}$$\right] B\left[$$\begin{gathered} \alpha \ -2 \alpha \ \alpha \end{gathered}$$\right]=\left[$$\begin{array}{lll} 4-8 & 4 \end{array}$$\right]\left[$$\begin{array}{lll} 1 & 3 & 4 \ 1 & 2 & 3 \ 4 & 4 & 4 \end{array}$$\right]\left[$$\begin{gathered} 4 \ -8 \ 4 \end{gathered}$$\right]} \\
\& =\left[$$\begin{array}{lll} 12 & 12 & 8 \end{array}$$\right]\left[$$\begin{gathered} 4 \ -8 \ 4 \end{gathered}$$\right] \\
\& = {48-96+32=-16}

\]

Solutions · 1