Answer to Question #289811 in Linear Algebra for Febr22

Question

Find A if (A^-1-3I)^T= 2 [-1 2

5 4]

Answer 1

Solution:

Given,

$(A^{-1}-3I)^T=2\begin{bmatrix} -1 & 2\\5 &4\end{bmatrix} \\ \Rightarrow (A^{-1})^T-(3I)^T=\begin{bmatrix} -2 & 4\\10 &8\end{bmatrix} \\ \Rightarrow (A^T)^{-1}-3I=\begin{bmatrix} -2 & 4\\10 &8\end{bmatrix} \\ \Rightarrow (A^T)^{-1}=\begin{bmatrix} -2 & 4\\10 &8\end{bmatrix}+3I \\ \Rightarrow (A^T)^{-1}=\begin{bmatrix} -2 & 4\\10 &8\end{bmatrix}+\begin{bmatrix} 3 & 0\\0 &3\end{bmatrix} \\ \Rightarrow (A^T)^{-1}=\begin{bmatrix} 1 & 4\\10 &11\end{bmatrix} \\ \Rightarrow (A^T)=\begin{bmatrix} 1 & 4\\10 &11\end{bmatrix}^{-1} \\ \Rightarrow (A^T)=\dfrac{1}{11-40}\begin{bmatrix} 11 & -4\\-10 &1\end{bmatrix} \\ \Rightarrow (A^T)=-\dfrac{1}{29}\begin{bmatrix} 11 & -4\\-10 &1\end{bmatrix} \\ \Rightarrow A=-\dfrac{1}{29}\begin{bmatrix} 11 & -4\\-10 &1\end{bmatrix}^T \\ \Rightarrow A=-\dfrac{1}{29}\begin{bmatrix} 11 & -10\\-4 &1\end{bmatrix}$