Question 52 of 472MediumMCQ
Let B=[1135] and A be a 2×2 matrix such that AB−1=A−1. If BCB−1=A and C4+αC2+βI=O, then 2β−α is equal to Solutions · 1
Official solution
\[\begin{aligned}
& B=\left[\[\begin{array}{|ccc|}1 & 3 \\ 1 & 5\end{array}\end{aligned}\]\right] \\
\& A B\^{-1}=A\^{-1} \\
\& \Rightarrow A\^2=B
\]
Also, BCB−1=A
\[⇒C⇒C4⇒C4=B−1AB=(B−1AB)(B−1AB)(B−1AB)(B−1AB)=B−1A4B=B−1B2B=B2\]
Also, \[C^2=\left(B^{-1} A B\right)\left(B^{-1} A B\right)\]
\[=B−1A2B=B−1BB\]
\[\begin{aligned}
& \Rightarrow C^2=B \\
& \Rightarrow C^4+\alpha C^2+\beta I=0 \\
& \Rightarrow B^2+\alpha B+\beta I=0 \\
& B^2=\left[\[\begin{array}{|ccc|}1 & 3 \\ 1 & 5\end{array}\end{aligned}\]\right]\left[\[1135\]\right]=\left[\[461828\]\right] \\
\& \Rightarrow\left[\[461828\]\right]+\alpha\left[\[1135\]\right]+\left[\[β00β\]\right]=\left[\[0000\]\right] \\
\& \Rightarrow 4+\alpha+\beta=0
\]
and \[18+3 \alpha=0\]
\[⇒α=−6⇒β=2⇒2β−α=10\] Answer the question to unlock