Given matrix $A$  \begin{bmatrix}  a_1b_1+1 &amp;a_1b_2 + 1&amp;...&amp;a_1b_n + 1\\ a_2b_1+1 &amp;a_2b_2 + 1&amp;...&amp;a_2b_n + 1\\ ...&amp;...&amp;...&amp;...\\ a_nb_1+1 &amp;a_nb_2 + 1&amp;...&amp;a_nb_n + 1 \end{bmatrix} Find its determinant.  I was using some properties of $\det$ here.  Suppose we have an arbitrary quadratic $n \times n$ matrix $A$.  Let's denote $A + \alpha$ matrix, where we added $\alpha$ to all elements of matrix $A$. Then $\det(A + \alpha) = \det(A) + \alpha \cdot\sum \limits_{i = 1}^n \sum \limits_{j = 1}^n A_{ij}$, where $A_{ij} = (-1)^{i + j}M_{ij}$.  So I got:  \begin{align} &amp; \det \begin{bmatrix}  a_1b_1+1 &amp;a_1b_2 + 1&amp;...&amp;a_1b_n + 1\\ a_2b_1+1 &amp;a_2b_2 + 1&amp;...&amp;a_2b_n + 1\\ ...&amp;...&amp;...&amp;...\\ a_nb_1 +1 &amp;a_nb_2 + 1&amp;...&amp;a_nb_n + 1 \end{bmatrix} \\ &amp; =  \det \begin{bmatrix}  a_1b_1 &amp;a_1b_2 &amp;...&amp;a_1b_n\\ a_2b_1 &amp;a_2b_2 &amp;...&amp;a_2b_n \\ ...&amp;...&amp;...&amp;...\\ a_nb_1 &amp;a_nb_2 &amp;...&amp;a_nb_n \end{bmatrix} + 1\cdot \sum \limits_{i = 1}^n \sum \limits_{j = 1}^n A_{ij}\ \ \ \ (*), \end{align} where $A$ is matrix without $+ 1$'s.  Then $(*) =  \det \begin{bmatrix}  b_1 &amp;b_2 &amp;...&amp;b_n\\ b_1 &amp;b_2 &amp;...&amp;b_n \\ ...&amp;...&amp;...&amp;...\\ b_1 &amp;b_2 &amp;...&amp;b_n \end{bmatrix} \cdot \prod \limits_{i = 1}^n a_i +  \sum \limits_{i = 1}^n \sum \limits_{j = 1}^n A_{ij} = 0 \cdot  \prod \limits_{i = 1}^n a_i + 0 = 0$, because we have equal strings everywhere.  So, $\det$ of initial matrix is equal to $0$.  Can you please check my result?
How to evaluate  $$\det\begin{bmatrix}1+x_1y_1&amp;\cdots &amp; 1+x_1y_n\\ \vdots &amp; \ddots &amp; \vdots \\ 1+x_ny_1 &amp; \cdots &amp; 1+x_ny_n\end{bmatrix}$$?  I tried factorisation but I have no idea. I could not factor anything out... I need a hint to kick off.   Edit: A conjecture due to multilinearity: if we let $v_j=(x_1y_j,\cdots,x_ny_j)^T$, then my guess would be $$\det\begin{bmatrix}1+x_1y_1&amp;\cdots &amp; 1+x_1y_n\\ \vdots &amp; \ddots &amp; \vdots \\ 1+x_ny_1 &amp; \cdots &amp; 1+x_ny_n\end{bmatrix}=\det(1,v_2,\cdots,v_n)+\det(v_1,1,\cdots,v_n)+\cdots+\det(v_1,v_2,\cdots,1)=\color{red}{0}$$ where $1$ denote $(1,\cdots,1)^T\in \Bbb K^n$,$\Bbb K$ the underlying number field.
I want to use a Java file SaveProp which is written in a package user. The class file has been placed in WEBINF/classes. Below are the two lines which cause the issue.  jsp:useBean id="user" class="user.SaveProp" scope="session"  jsp:setProperty name="user" property="*"   It throws:  The value for the useBean class attribute user.SaveProp is invalid   I'm using Tomcat 6.0. Looking for the actual reason of refraction explained precisely without analogies Why do objects sometimes fall on their own? In the Harry Potter universe, wizards and witches have managed to keep Muggles unaware of their existence, across the planet and over many centuries. Up to the timeframe that the novels take place in (the 1990s), I can see how this can be achieved through a combination of discretion and magic, with discretion being the preferred method on occasion (e.g., you don't need magic to slip into Platform 9 3/4 unnoticed, you just need to check that nobody is looking).   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