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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Alternate formula for the determinant of a 3x3 Matrix

So, I've recently been reading about Dodgson condensation, for matrices. Take, in particular the $3x3$ case. Define a matrix, $|A|:$ \begin{pmatrix} a & b & c \\ d &...
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What is an intuition behind permanent?

I would like to know what is your intuition behind permanent of a matrix. For me, it looks like someone came and saw determinant, deleted permutation sign and voila, we have permanent and it counts ...
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Finding Factors of a Determinant

Consider the determinant with elements: $a_{11} = ax-by-cz, a_{12}=ay+cz, a_{13}=cx+az$ $a_{21}=ay+bx, a_{22}=by-cz-ax, a_{23}=bz+cy$ $a_{31}=cx+az, a_{32}=bz+cy, a_{33}=cz-ax-by$ Where $a_{ij}$ ...
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On the definition of the volume form in general vector spaces as given in Spivak, Calculus on Manifolds

For a vector space $V$ denote by $\Lambda^k(V)$ the space of alternating $k$-tensors, or alternating $k$-fold multilinear maps on $V$. I have some difficulty following the intention of the author in ...
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Application of the Leibniz formula

I am trying to show that in the characteristic polynomial of $\det (A-\lambda I)$ the coefficient of $\lambda^{n-1}$ is $(-1)^{n-1}TrA.$ I'm given the hint to use the Leibniz formula for this ...
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How to derive the Vandermonde Determinant?

I watched this video https://www.youtube.com/watch?v=87iJTcXqTKY explaning the Vandermonde Determinant I understood everything but I was wondering why the guy never mentioned the (-1)^(i+j) term used ...
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How to prove that a square matrix's determinant is zero given its row and column properties? [duplicate]

How can I prove the following? If $A$ is an $n \times n$ matrix such that $$\sum\limits_{j=1}^n a_{ij} = 0$$ for $1 \leq i \leq n$ then $\det A = 0$.
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Prove that the determinant is a multiple of $17$ without developing it

Let, matrix is given as : $$D=\begin{bmatrix} 1 & 1 & 9 \\ 1 & 8 & 7 \\ 1 & 5 & 3\end{bmatrix}$$ Prove that the determinant is a multiple of $17$ without developing it? ...
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A different determinant expansion

Let us denote the determinant of a matrix $A=((a_{ij}))_{i,j=1}^n$ as $\displaystyle \det_{i,j=1}^n a_{ij}$. Let $\delta_{ij}$ be the kronecker delta function and suppose $a_{ij}=a_{ji}$ for all ...