# Tagged Questions

Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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### How to prove a Wronskian identity?

The following Wronskian identity can be proved by expanding both sides and checking that two sides are the same. But how to prove it more elegantly? Let $u_1(x), u_2(x), u_3(x), u_4(x)$ be four ...
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### Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?

I saw the following statement in my homework and we are asked to make use of the statement: If $S$ is a symmetric matrix then $$\det(I + S ) = 1 + \operatorname{trace}(S).$$ However, I am not ...
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### Show determinant of matrix is non-zero

I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help please....
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### Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$

I have to prove/disprove this: If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$ I believe it is true but I can not think of a direct way to prove it. Any ...
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### The “second derivative test” for $f(x,y)$

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
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### Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?

Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
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### How do I prove that $\det A= \det A^t$?

I found this exercise in Artin. It asks me to prove that $\det A= \det A^t$ where $A^t$ is the transpose of the matrix $A$. Can anyone please comment whether my proof is correct or not? Attempted ...
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### Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its determinant,...
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### Limit of a sequence of determinants.

Let $\beta>0$ be given. For each $n\geq 2$, let $\Delta_n=\det M_n$ denote the determinant of the following matrix: \begin{align} M_n = \begin{pmatrix} 2+\epsilon^2 & -1 & 0 & 0 &...