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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1answer
17 views

Sum-of-divisors determinant

Let $\sigma_k(n)=\sum_{d|n}d^k$ be the generalized sum-of-divisors function. Let $S_n$ be the matrix defined by $[S_n]_{ij}=\sigma_i(j)$. I read a comment somewhere that $$\det(S_n)=1!\cdot 2!\cdots ...
2
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1answer
15 views

Proving Cayley formula using Kirchhoff matrix theorem?

To count the number of spanning trees of a complete graph of order $n$ one can use Kirchhoff matrix theorem and arrive at the exact answer $n^{n-2}$. But in doing so, one should know how to evaluate ...
-1
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1answer
20 views

Proving multilinearity of determinant [on hold]

As the title says, how we can prove multilinearity property of determinants: $$ \begin{vmatrix} p+q+r & x+y+z & u+v+w \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\\ ...
2
votes
1answer
38 views

Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...
1
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0answers
21 views

Unknown functions yield a given determinant

I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$. This amounts to finding twelve smooth ...
2
votes
3answers
105 views

How can we memorize the formula for the determinant of a $4\times4$ matrix?

This is the formula for the determinant of a $4\times4$ matrix. . 0,0 | 1,0 | 2,0 | 3,0 0,1 | 1,1 | 2,1 | 3,1 0,2 | 1,2 | 2,2 | 3,2 0,3 | 1,3 | 2,3 | 3,3 . ...
1
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3answers
57 views

How do I find the determinants of $3A, -A, A^2, A^{-1}$, where A is an $4\times 4$ matrix and $\det(A) = \frac{1}{3}$?

I am getting crazy with these determinants. For a little, I thought I could solve a problem alone, because I had understood more or less how to calculate the determinants of a matrix, but I am back to ...
5
votes
4answers
346 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
3
votes
1answer
50 views

Verifying whether a number is the determinant of a matrix

What is the (computationally) fastest way to determine whether a number is the determinant of a given real matrix? I am wondering if I have an upper bound on the absolute value of the determinant of ...
1
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0answers
18 views

How to find the determinant of a parallelogram using the vertices. (Using a matrix)

What it says on the tin. I already know how to find the determinant of a parallelogram using the vector components in a matrix, however, I am curious if there is a way to do it simply through the ...
9
votes
3answers
319 views

Calculate a determinant.

Let $a_{1}, \cdots, a_{n}$ and $b$ be real numbers. I like to know the determinant of the matrix $$\det\begin{pmatrix} a_{1}+b & b & \cdots & b \\ b & a_{2}+b & \cdots & b ...
1
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0answers
25 views

Can the given transformation possible for given determinant?

In forth step $(x-1)(x-2)$ is obtained by applying transformation R$1 \frac{1}{(x+1)}$ and R$2 \frac{1}{(x+2)}$. But we get value of $x = -1$ or $ x = -2$ so $\frac{1}{(x+1)}$ and ...
3
votes
1answer
28 views

Evaulate a determinant involving factorials.

In a problem set given by a teacher, there is the following problem. If $a_n = \frac{1}{n!}$, evaluate $$ D_n = \begin{vmatrix} a_1 & a_0 & 0 & 0 & \cdots & 0 & 0 & ...
0
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0answers
15 views

Determinant expansion about rows

Here is a theorem from Beezer's A First Course in Linear Algebra: I am having a hard time understanding why there is a $-1$ in $i - 1 + l - \epsilon_{lj}$, could anyone explain?
4
votes
2answers
62 views

If $BA$ has $-1$ as an eigenvalue, then so does $AB$?

I was just encountered with a rather tough problem as follows: Suppose $A,B\in M_n(\mathbb R)$, prove: $$\det(I_n+AB)\ne0\Rightarrow\det(I_n+BA)\ne0$$ Although at this moment I am still at a ...
8
votes
3answers
1k views

Determinant of a 5 × 5 matrix

I have a little problem with a determinant. Let $A = (a_{ij}) \in \mathbb{R}^{(n, n)}, n \ge 4$ with $$a_{ij} = \begin{cases} x \quad \mbox{for } \,i = 2, \,\, j \ge 4,\\ d \quad \mbox{for } ...
5
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1answer
67 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
0
votes
1answer
40 views

Tutte matrix - Determinant

I'm trying to understand the proof of the "magic theorem" about the Tutte matrix which states: Let $T$ be the Tutte matrix of $G(V, E)$. Then, $$\det(T) = 0 \quad\Longleftrightarrow\quad G ...
1
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1answer
38 views

Finding determinant of a 4x4 matrix

I am trying to find the determinant of this matrix but was told by my teacher that we wouldn't need to find the determinant of more than $3\times 3$ matrices so I am guessing there is a way of solving ...
0
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1answer
14 views

Determinant with one parameter, how to deal with this?

Let $t\in \mathbb R$ be a parameter, and $$|A(t)|= \begin{vmatrix} a_{11}+t &a_{12}+t &\cdots &a_{1n}+t\\ a_{21}+t &a_{22}+t &\cdots &a_{2n}+t\\ \vdots &\vdots ...
0
votes
2answers
41 views

Finding complex eigenvalues

For the matrix \begin{pmatrix}1/2 & 1 & 3/4\\2/3 & 0 & 0\\0 & 1/3 & 0\end{pmatrix} Find the eigenvalues and corresponding eigenvectors. I did this with an online calculator and ...
0
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3answers
37 views

Why would the Jacobian not be zero in this case?

Find the jacobian of the transformation x = u, y = 3uv in the uv plane. Why would $U_y$ not be zero in this case, if the equation U = x contains no mentions of y?
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0answers
21 views

How to find spectral radius of ${0,1}$ and ${0,1,-1}$ matrices?

[this is kind of a continuation of this question ] It seems that the following is true, Among $n=3$ dimension symmetric matrices over $\{0,1\}$ which have $d=7$ ones the maximum spectral radius is ...
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2answers
63 views

An (open?) problem about a sequence of nested sub-matrices and their determinant

I had an idea. Let us start with an example. Consider the matrix $$ A = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] $$ It is invertible, ...
0
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2answers
49 views

Process of finding the eigenvalues of a 3x3 matrix

I'm trying to find the eigenvalues of a 3x3 matrix in order to eventually find an orthogonal matrix $Q$ and diagonal matrix $D$ such that $Q^TAQ = D$, where $A$ is a symmetric matrix, however I'm not ...
6
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4answers
183 views

Any hint about solving this monster determinant?

I'm asked to solve the following determinant: $$|A|= \begin{vmatrix} 1 &2 &3 &\cdots &{n-1} &n\\ 2 &3 &4 &\cdots &n &1\\ \vdots &\vdots &\vdots & ...
1
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2answers
32 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
0
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1answer
44 views

What is the determinant of the sum of a diagonal matrix and a matrix of ones?

Given a square matrix, all elements outside of the main diagonal being equal to $1,$ what is its determinant?
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4 views

Link between the cofactors of two related symmetric positive-definite matrices

Let $S = \left( s_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric positive-definite matrix. Let $\Sigma = \left( \sigma_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric ...
10
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2answers
121 views

Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)

In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...
2
votes
2answers
70 views

How do you find the determinant of this $(n-1)\times (n-1)$ matrix?

It's for a proof of Cayley's Formula, I know I'm being dumb and can't see it, how do I find the determinant of this $(n-1)\times (n-1)$ matrix where the diagonal entries are $n-1$ and the off diagonal ...
2
votes
0answers
39 views

Value of determinant using given conditions.

Let $A$ be a $2$ x $2$ matrix with real entries and $det(A)$ is equal to $d$ which is non-zero. It is given that $det(A +d(adjA))=0$ where $adj$ stands for the adjoint of the matrix. We have to find ...
0
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0answers
15 views

Computing the spherical coordinates in n-dimensions

This time I want to compute the Jacobian of the spherical coordinates in n dimensions, so it needs to give me the following result: $$\displaystyle ...
1
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2answers
58 views

How to calculate the determinants like these?

I'm trying to solve this determinant question and I just can't understand how to approach this. If $x^3$=1, then $$\Delta=\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b ...
0
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0answers
17 views

Problems with the inverse of a banded matrix: not invertible?

I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is ...
0
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1answer
27 views

Determinant using Row and Column operations/expansions

We are asked to show that: $$ \det\left[\begin{array}{rrr} 2 & 3 & 7 & 1 & 3\\ 2 & 3 & 7 & 1 & 5\\ 2 & 3 & 6 & 1 & 9\\ 4 & 6 ...
2
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0answers
18 views

How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
2
votes
3answers
35 views

How to explain the calculation of the determinant of a $4\times4$ matrix

In my linear algebra lecture notes, I am studying an example which concerns the calculation of the determinant of a $4 \times 4$ matrix, by first reducing the matrix to upper triangular form. (See ...
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0answers
20 views

Determinant of block-triangular matrix made of 3 matrices [duplicate]

Let $A$ be a $k \times k$ matrix and $B$ be a $\left(n-k\right) \times \left(n-k\right)$ matrix, and $Z$ be the $n \times n$ matrix $$ Z = \left( \begin{matrix} A & C \\ 0 & B ...
1
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2answers
32 views

Determinants order of operations

When computing determinants using their properties, what is the order in which the determinant gets evaluated? Ie. \begin{vmatrix} 2AA^t \\ \end{vmatrix} Do we start with $2A$ or ...
2
votes
2answers
28 views

Computing the volume of a fundamental domain of a lattice

Suppose I have $n$ linearly independent vectors in $\mathbb{R}^m$, say $v_1, .., v_n$. Then $v_1,..., v_n$ form a lattice $\Lambda$ inside a subspace $V$ = $\mathbb{R}v_1 + ... + \mathbb{R}v_n ...
1
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1answer
28 views

Why is determinant called volume of the fundamental parallelepiped in geometry of numbers?

Let $v_1, ..., v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$. Then they form a lattice $\Lambda \subseteq \mathbb{R}^n$ and the volume of the fundamental domain is $|\det A|$, where $A$ ...
1
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0answers
45 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
0
votes
3answers
33 views

Find determinant value

\begin{vmatrix} 3 & 2 & 0 & 0 & . &. & . & . &0 &0 \\ 1 & 3 & 2 & 0 & . &. & . & . &0 &0 \\ 0 & 1 & 3 & 2 & . ...
0
votes
0answers
29 views

The discriminant of (1,x,x^2) in a cubic field.

Let $K$ be a cubic field such that $K=\mathbb Q[x]$ with $x^3=2$. The discriminant of $\{1,x,x^2\}$ is supposed to be $\begin{vmatrix} 3 & 0 & 0 \\ 0 & 0 & 6 \\ 0 & 6 & ...
3
votes
2answers
56 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
0
votes
2answers
42 views

Same roots, same polynomial? How to prove characteristic polynomial of $AB = BA$?

I'm giving a (simple) proof that the characteristic polynomial of $AB$ = characteristic polynomial of $BA$ (without using the fact that $AB$ and $BA$ are similar). $det(AB) = det(A)det(B) = ...
5
votes
3answers
84 views

Prove that $\det(AA^T+I)\ge 1$

If $A$ is a matrix with real entries, prove that $$\det(AA^T+I)\ge 1.$$ I tried using the eigenvalues. One thing came into my mind: maybe $AA^T$ is positive definite (I don't know whether this is ...
0
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0answers
14 views

How do we sketch the ellipse determined by $T(\vec{x})$ and determine its axes, given an expansion factor?

I have been told that if $\left\{\exists \, T(\vec{x})^{-1}\mid T(\vec{x})=A\vec{x} \mid \mathbb{R}^2\mapsto\mathbb{R^2}\right\}$, then the image $T(\Omega)$ of the unit circle $\Omega$ is an ellipse. ...
2
votes
4answers
77 views

An Eigen Value of $\tiny \begin{pmatrix} a&b&1 \\ c&d&1 \\ 1&-1&0\\ \end{pmatrix}$ is :

Let $a,b,c,d$ be distinct non zero real numbers with $a+b=c+d.$ Then, an eigen value of the matrix $A= \begin{pmatrix} a&b&1 \\ c&d&1 \\ 1&-1&0\\ \end{pmatrix}$ is : $(i)~a+c ...