Questions tagged [determinant]
Questions about determinants: their computation or their 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.
6,963
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Polynomial factorization using determinant
Algebraic identities
$$
P_2(a,b)=a^3+b^3=(a+b)(a^2+b^2-ab)
$$
$$
P_3(a,b,c)=a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)
$$
$$
P_4(a,b,c,d)=a^3+b^3+c^3+d^3-3abc-3abd-3acd-3bcd=(a+b+c+d)(a^2+b^2+c^2+...
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Proof of transposed matrix related equation with Levi-Civita symbol
I would like to prove the following equation in relation to Levi-Civita symbol but I got stuck.
$\left|{A^T}\right|=\left|{A}\right|$ (1)
where A is a square matrix of size 3.
I am also given the ...
- 1
2
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1
answer
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Is the approach I did to justify this limit equality correct?
I am reading a solution to a problem and got stuck to understand the equality (if $A(t)$ is an invertible matrix whose entries depend in a differentiable way on a parameter $t$ and $ \lim_{h\to0}P(h)=...
- 183
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dividing the determinant of a matrix by the expected value of the determinant of the same matrix over a uniform distribution
Let $A, B$ be square $n \times n$ matrices as follows:
$$
A = \begin{bmatrix}
x_1&x_2&\cdots&x_n\\
x_{n+1}&x_{n+2}&\cdots&x_{2n}\\
\vdots&\vdots&\vdots&\vdots\...
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3
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Binomial determinant and LU decomposition
Let $A_n$ be following $n \times n$ symmetric pentadiagonal matrix
$$
\begin{pmatrix}
6&4&1&& \\
4&\ddots&\ddots&\ddots&\\
1&\ddots&\ddots&\ddots&1\\
&...
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Is this generalization of determinant for a higher-order tensor a standard object?
The determinant of an $n$ by $n$ matrix $a$ can be defined as
$$ \mathrm{det}(a)= \sum_{\sigma} \mathrm{sgn}(\sigma) a_{1,\sigma(1)} a_{2,\sigma(2)} \dots a_{n,\sigma(n)}$$
where $\sigma$ is a ...
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Bound on determinant when replacing columns
Suppose a square matrix $A$ has columns $a_1, \ldots, a_n$.
Construct $A'$ by replacing a column $a_k$ with $a_k'$ such that $||a_k - a_k'|| \le \epsilon$. Is it possible to bound $|\det A - \det A'|$?...
- 321
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Given a map, is the determinant preserved regardless of representation?
I was exploring linear algebra a bit, and I stumbled upon a question that is now eating at me. Given a map, is the determinant preserved regardless of representation? Geometrically, I understand that ...
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Proof about multiplying another submatrix while calculating the determinant [duplicate]
I took a linear algebra class. However, while lecturing on inverse matrix, the professor said that the following content is trivial, so We skipped over the proof. However, I was so curious about the ...
- 393
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1
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Create a matrix that has a definite determinant
Consider I want to create a matrix of order $3\times 3$ that has a determinant of $25$. How can I accomplish that task? Even if I have a computer program for that, what shall be range that I need to ...
- 134
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1
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50
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Determinant of $3\times3$ block matrix [closed]
My intuition tells me that this determinant should be relatively simple to compute, yet after a while trying, I am running out of ideas. So any suggestions would be greatly appreciated.
We aim to find ...
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4
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To find the trace and determinant of a matrix $A$ satisfying $A^{2023} + A = \left(\begin{smallmatrix}2 &2& 0\\ 0&2&2\\ 0&0&2\end{smallmatrix}\right)$ [closed]
If $A$ is a $3 \times 3$ matrix such that
$$
A^{2023} + A = B\quad \mbox{where matrix}\ B\
\mbox{is given by}\quad B =
\left(\begin{smallmatrix}2 &2& 0\\ 0&2&2\\ 0&0&2\end{...
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Proof of inequality involving matrices: $\operatorname{tr}(I-\Lambda) + \log \det \Lambda \leq 0$
Notation
Let $I_n$ be an $n$th order unit matrix.
Problem
We want to show that
$$
\operatorname{tr}(I_n-\Lambda) + \log \det \Lambda \leq 0,
$$
where $n\times n$ matrix $\Lambda$ is a positive ...
- 590
1
vote
1
answer
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Why do we generally don't expand a determinant diagonally? Why do we generally expand a determinant either row wise or column wise?
Q)Why do we generally don't expand a determinant diagonally? Why do we generally expand a determinant either row wise or column wise ?
I learnt about matrices and determinant in class 12. Generally ...
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1
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Finding All Linear Factors of a Determinant for a Given 3x3 matrix
I am working on finding all the linear factors of the determinant of the following $3\times 3$ matrix:
\begin{vmatrix}
a & a^3 & a^4 \\
b & b^3 & b^4 \\
c & c^3 & c^4
\end{...
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