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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5
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1answer
61 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
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1answer
27 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
vote
1answer
35 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
votes
1answer
13 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
36 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
34 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?
0
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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 ...
1
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2answers
62 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
votes
2answers
45 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 ...
5
votes
4answers
182 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
vote
2answers
31 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
votes
1answer
42 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?
0
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0answers
3 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
votes
2answers
119 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
69 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
36 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
54 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
16 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
34 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
31 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
27 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
44 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 ...
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0answers
29 views

How to solve this determinant problem? [closed]

If $A$ is a $3\times3$ matrix and $|A|=2$, find $$\lvert A^{-1}+4\mbox{adj}(A)\rvert$$ Thank you very much!
0
votes
3answers
31 views

Find determinant value

\begin{vmatrix} 3 & 2 & 0 & 0 & . &. & . & . &0 &0 \\ 1 & 3 & 2 & 0 & . &. & . & . &0 &0 \\ 0 & 1 & 3 & 2 & . ...
1
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0answers
27 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
53 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
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2answers
41 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
82 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
75 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 ...
2
votes
1answer
55 views

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are ...
1
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1answer
52 views

Is the area of $T(\Omega)=|\det A|\,(\text{area of }\Omega)$?

We are given that $\Omega$ is a parallelogram in $\mathbb{R}^3$ and $\left\{ T(\vec{x}) = A\vec{x} \mid \mathbb{R}^3 \mapsto \mathbb{R}^3\right\}$ is a linear transformation. From the definition of ...
2
votes
1answer
71 views

What is the 3-volume of the 3-parallelepiped defined by $\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}$?

We have $\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}=\left\{\begin{bmatrix}1\\0\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\2\\3\\4\end{bmatrix}\right\}$ ...
0
votes
1answer
34 views

can we use saras method for finding determinant of matrix greater than 3x3

Can Sarrus method of finding determinant be used for finding determinant of matrices greater than $3\times 3$ http://en.wikipedia.org/wiki/Rule_of_Sarrus as I am unable to find any example of matrix ...
1
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0answers
43 views

Induction proof: $\det(M) = \prod_{1 \le j \le n} (x_j - x_i)$

Following problem: Let $\mathbb{K}$ be a Field and $M = \begin{pmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ 1 & x_n & ...
0
votes
2answers
26 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...
0
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0answers
8 views

fix a line for operation on matrices

operations on matricies determinant Why does a line at least must be fixed to to operations on matrices for their determinant calculation? I understand that if it was allowed not to do such a ...
0
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0answers
25 views

$j$-volume of $j$ dimensional parallelepiped inside $\mathbb{R}^n$

Let $v_1, ..., v_j \in \mathbb{R}^n$ be linearly independent. Let $V = \mathbb{R}v_1 + ... + \mathbb{R} v_j$ be a subspace of $\mathbb{R}^n$ and $\Gamma = \mathbb{Z}v_1 + ... + \mathbb{Z} v_j$ a ...
3
votes
3answers
40 views

Determinant of the sum of some special matrix

$A,B$ are $3\times 3$ matrices. It is known that: $\det(A)=0$ $\forall i,j: b_{ij}=1$, where $b_{ij}$ is an element of matrix $B$ $\det(A+B)=1$ Find $\det(A+2014B)$ I don't know what to do. I ...
0
votes
1answer
38 views

Matrix derivatives of determinant and inverse related to $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$

I would like to calculate the derivatives of determinant and inverse related to the term $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$ with respect to $\mathbf{X}$, where $\mathbf{C}$ is a constant matrix. ...
1
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1answer
52 views

is it possible to use induction to prove the following?

I know for sure that there is some easy way to prove what I am about to tell, but, at first, I'd like to know if I can set up a proof by induction for two "cross-referenced formulas". I have two ...
-1
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3answers
47 views

How do you prove (or disprove) the statement: If $A^3 = 0$, then A-I is non-singular

I've proven something similar: A*A =0, then A + I is non-singular for 2x2 matrices. But not sure how to proceed for $A^3 = 0$, then A-I is non-singular Also, not sure how to prove A*A =0, then A + I ...
3
votes
1answer
86 views

Inquiry about determinant of $ \left(\begin{matrix} A & B \\ B^T & C \end{matrix}\right)$

Based off of http://en.wikipedia.org/wiki/Determinant#Block_matrices, I'm trying to find the formula for $\det(M)$ when $M = \left(\begin{matrix} A & B \\ B^T & C \end{matrix}\right)$. It is ...
0
votes
3answers
32 views

Counter example to the converse of the special property of triangular matrices

Let $A = (a_{ij})$ be a square matrix of order $n$ over $\mathbb R$. Give a counter-example of: $det(A) = \prod_{t = 1}^n a_{tt}$ $\implies$ $A$ is a triangular matrix. Any help? Thanks.
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0answers
41 views

Area of parallelogram from a linear operator

let $L: \mathbb R^2\to\mathbb R^2$ be a linear operator which is invertible. Let {$u, v$} be a linearly independent set in $\mathbb R^2$. Find a formula for the area of the parallelogram induced by ...