Question

If $e\begin{bmatrix}e^{x}&e^{y}\\ e^{y}&e^{x}\end{bmatrix} = \begin{bmatrix}1&1\\ 1&1\end{bmatrix}$, then the values of $x$ and $y$ are respectively

• A. -1,-1
• B. 1, 1
• C. 0, 1
• D. 1, 0
• E. 0, 0

Answer 1

Answer: (A)

Given, $\begin{bmatrix}e^{x} & e^{y} \\ e^{y} & e^{x}\end{bmatrix}]=[\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}e^{x+1} & e^{y+1} \\ e^{y+1} & e^{x+1}\end{bmatrix}=\begin{bmatrix}{cc}e^{0} & e^{0} \\ e^{0} & e^{0}\end{bmatrix} (\because e^{0}=1)$
On equating the corresponding elements, we get
$e^{x+1}=e^{0} $ and $e^{y}+1=e^{0}$
$\Rightarrow x+1=0 $ and $y+1=0$
$\Rightarrow x=-1$ and $y=-1$