Deepak Sarma's profile - activity

I have a matrix A of order $n\times n$. Now I need another matrix B whose (i,j)th entry is $det A(i,j)$, where $det A(i,j)$ denote the determinant of the sub matrix of A formed by deleting $i^{th}$ and $j^{th}$ rows and columns. I'm unable to generate the submatrices $A(i,j)$ for every element at a time.

I have some functions, all of which are functions of variable $x$ but some of them may not have any positive solutions. It is known that at least one of them have a positive solution. Now I need a list of all smallest positive solutions for those functions. For example consider $f=x^2+3x+2$ and $g=2^{(5x + 1)} - 3.2^{(3x + 1)}$. Here $f$ doesn't have any positive root but $g$ has (0.792481250360578). I want a sage code like min(solve([f,x>1],x))+min(solve([g,x>1],x)) to get the list as [0.792481250360578]. Thank you in advance.

Thank you for your solution. It works with that particular matrix. But it doesn't work for all matrices.

For example consider $$A=\left(\begin{array}{rrrrr} 0 & 2^{x} & 1 & 1 & 1 \ 2^{x} & 0 & 1 & 1 & 1 \ 1 & 1 & 0 & 2^{x} & 2^{x} \ 1 & 1 & 2^{x} & 0 & 2^{x} \ 1 & 1 & 2^{x} & 2^{x} & 0 \end{array}\right)$$

or

$$A=\left(\begin{array}{rrrr} 0 & 1 & 2^{x} & 1 \ 1 & 0 & 1 & 2^{x} \ 2^{x} & 1 & 0 & 1 \ 1 & 2^{x} & 1 & 0 \end{array}\right)$$

Here I get an error "cannot evaluate symbolic expression numerically"

Note: For all my matrices, it is known that the solution is at most equal to 2. So if something like min(find_root(X,0,2)+find_root(Y,0,2)) could be used, that is also enough for me.

Let us consider the $4\times 4$ symmetric matrix $$ A_x=\left(\begin{array}{rrrr} 0 & 1 & 1 & 1 \ 1 & 0 & 2^x & 2^x \ 1 & 2^x & 0 & 2^x \ 1 & 2^x & 2^x & 0 \end{array}\right) $$

Here I need to find $\min { x>0: det(A_x)=0 \, or \, ||A_x^{-1}||=0 } ,$ where by $||M||$ we mean the sum of all entries of the matrix $M.$ I'm looking for a general sage program where my input will be a matrix with entries as functions of an inderminant (like the matrix $A_x$ above) which will give me the unique $x$ corresponding to my matrix. If no such real value exists, it should result as $\infty$ Can anyone help me? Thank you in advance.

Let us consider the $4\times 4$ symmetric matrix $$ A_x=\left(\begin{array}{rrrr} 0 & 1 & 1 & 1 \\ 1 & 0 & 2^x & 2^x \\ 1 & 2^x & 0 & 2^x \\ 1 & 2^x & 2^x & 0 \end{array}\right) $$

Here I need to find $\min { x>0: det(A_x)=0 \, or \, ||A_x^{-1}||=0 } ,$ where by $||M||$ we mean the sum of all entries of the matrix $M.$ If anyone helps me solving this problem using sage, this will be enough for me. Thank you