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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2
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1answer
54 views

What can be said about functions of constant Hessian determinant?

Let $f:\mathbb{R}^2\to \mathbb{R}$ with $\det \nabla^2f = 1.$ Let's also assume that $\nabla^2 f$ is positive-definite (which we can do WLOG by adjusting the sign of $f$). What can we say about $f$? ...
0
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0answers
49 views

Strategy for simplifying 3x3 determinants?

Is there any general strategy for simplifying 3x3 determinants in the form of: \begin{vmatrix} a&b&c\\ b&d&e\\ c&e&f\\ \end{vmatrix} where a,b,c,d,e,f may not ...
0
votes
1answer
29 views

Angle between 2 vectors using the determinant

I have a polygon like this: I basically want to find the angles $\alpha$, inside the polygon, between the vectors. I'm using the determinant to calculate the angle alpha: $det(\vec V2, \vec V2 ) ...
0
votes
5answers
64 views

If $A =\begin{pmatrix} -1 & 0 & 1\\ 0 & 1 & 1\end{pmatrix}$ and $AB = I$ find the $3\times 2$ matrix $B$.

Alright so you multiply $A$ and $B$ and you get four equations. Then you do $\det[AB] = \det[I] = 1$ and you get a fifth. I'm stuck here now. What else can I do to find $B$? I'm trying to get this ...
1
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4answers
98 views

How to prove the following exercise by using the definition of a determinant?

$\begin{align} \begin{vmatrix} a_{11} & \cdots& a_{1m} & 0 & \cdots & 0 \\ \cdot & \cdots & \cdot & \cdot & \cdots & \cdot \\ a_{m1} & \cdots & a_{mm} ...
0
votes
1answer
49 views

How to find the determinant of a NxN matrix

Here is my matrix. How do I find the determinant of this one? I'm really trying to solve it but I can't think of anything. $$ \begin{pmatrix} 3 & 2& ...& 2\\ 2& 3& ...& 2\\ ...
3
votes
1answer
36 views

Determinant (or positive definiteness) of a Hankel matrix

I need to prove that the Hankel matrix given by $a_{ij}=\frac{1}{i+j}$ is positive definite. It turns out that it is a special case of the Cauchy matrices, and the determinant is given by the Cauchy ...
8
votes
3answers
85 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
2
votes
0answers
40 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
1
vote
1answer
17 views

Bound on the degree of a determinant of a polynomial matrix

I want to implement a modular algorithm for computing the determinant of a square Matrix with multivariate polynomials in $\mathbb{Z}$ as components (symbolically). My idea is first to reduce the ...
1
vote
2answers
59 views

Determinant of $U$, Determinant of $U^T$

Given an $n\times n$ matrix $U$ such that $U^TU = I_n$, the $n\times n$ identity matrix. Then what are the possible values of the determinant of $U$?
0
votes
0answers
45 views

The determinant of a matrix

In order get the determinant of$$\begin{pmatrix} \lambda-n-1 & 1 & 2 & 2 & 1 & 1 & 1& 1 & \cdots &1 & 1 \\ 1 & \lambda-2n+4 & 1 & 2 & 2 &2 ...
10
votes
2answers
133 views

Prove or disprove : $\det(A^k + B^k) \geq 0$

This question came from here. As the OP hasn't edited his question and I really want the answer, I'm adding my thoughts. Let $A, B$ be two real $n\times n$ matrices that commute and $\det(A + ...
1
vote
1answer
36 views

determinant of an endomorphism

Let $n \in \mathbb{N}$. We define: $f_n:\mathbb{R}[x]_{\leq n} \rightarrow \mathbb{R}[x]_{\leq n}, p\mapsto(p \cdot x)' $, where $q'$ is the derivation of a math. polynomial $q \in \mathbb{R}[x]$. ...
2
votes
2answers
58 views

How to calculate this determinant? [closed]

$F(x) = x(x - 1)(x - 2) \cdots (x - n + 1)$ good morning I want to prove that I want to calculate the determinant $$\left| \begin{array}{*{20}c} {F(a)} & {F'(a)} & {F''(a)} & \cdots ...
2
votes
1answer
41 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
1
vote
1answer
15 views

determinant of special structure block matrix

How do you compute the determinant of the block matrix: $$ M = \begin{bmatrix} A+B & A &A &A &... \\ A & A+B &A &A &... \\ A & A &A+B &A &... ...
1
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2answers
35 views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
0
votes
1answer
38 views

Jacobian matrix and determinant - relation to orientation

$F$ is a function from $V$ to $V$ where $V$ is a $n$-dimensional vetor space and $p \in V$. In the article Jacobian determinant it says: "If the Jacobian determinant at $p$ is positive, then $F$ ...
1
vote
1answer
46 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
1
vote
1answer
68 views

Does there exist $B$ for which $BB^T=I$?

My question is Does there exist a real matrix $B_{n\times m}$ with $m<n$ for which $BB^T=I_n$? Why do I need this? Suppose we are given a real matrix $Q_{m\times n}$ (again, with ...
4
votes
1answer
71 views

Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$

Let $M$ be a (real) $n \times n$ matrix. For $1 \leq i, j \leq n$ we denote by $M_{ij}$ the $(n-1) \times (n-1)$ matrix that we get when the $i$th row and $j$th column of $M$ are removed. Now, ...
1
vote
1answer
26 views

Determinant with Levi-Civita Symbol?

From Schaum's Outline in Tensor Calculus If $A = [a_{ij}]_{nn} $ is any square matrix, then define $\text{ det } A = \epsilon_{i_1i_2i_3...i_{n-1}i_n}a_{1 \, \cdot \, i_1}a_{2 \, \cdot \, ...
0
votes
2answers
25 views

Evaluating determinant of an implicit matrix

I know that row operations does not change the determinant of a matrix but I also know that for example, A is a nxn matrix and if det(A) = 2 then, det(2A) = (2^n)*det(A). So, how should I approach ...
0
votes
1answer
17 views

Matrices in systems of linear equations

I've been working on matrices lately. Currently, I am stuck on solving systems of linear equations using matrices. I've read the following article which has proved very helpful in understanding the ...
4
votes
1answer
69 views

Characterization of positive definite matrix with principal minors

A matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, for ...
2
votes
2answers
205 views

Why this representation of circle is valid?

A line passing through two distinct points $P_1(x_1,y_1),P_2(x_2,y_2)$ can be expressed by $$\det\left| \begin{array}{ccc} x-x_1&y-y_1 \\ x_2-x_1&y_2-y_1 \\ \end{array} \right|=0$$ Since ...
0
votes
4answers
42 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
6
votes
2answers
189 views

Determining the derivation of a determinant

Let $\Phi\colon E\to M$ with $E\subset \mathbb{R}\times M$ and $M\subset\mathbb{R}^n$ open. Consider the function given by $x\mapsto \Phi(t,x)$ for fixed $t\in\mathbb{R}$. (1) Determine $$ ...
0
votes
1answer
21 views

Proof regarding effect of row operations on determinants>

Let $A,B \in K^{n,n}$ and suppose $B$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$. Prove $det(A)=det(B)$. My Attempt I tried to use proof by induction for this . Take ...
1
vote
1answer
35 views

Find determinant $\det M$, where $m_{ij}=a_ia_j$, and $m_{ii}=a^2_i+k$

Let ${a_1,\dots,a_n}$ --- sequence and $k\ne 0$. Define matrix $M$ in following way: $m_{ij}=a_ia_j$ if $i\ne j$, and $m_{ii}=a^2_i+k$. Find $\det M$.
2
votes
1answer
52 views

Coordinate-free proof of determinant of transpose

I'm interested in a coordinate-free proof of the statement $\mathrm{det}(A) = \mathrm{det}(A^T).$ Let $V$ be a finite-dimensional vector space over a field $K$, and let $f : V \rightarrow V$ be an ...
7
votes
2answers
100 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
2
votes
1answer
39 views

If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
0
votes
2answers
62 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
12
votes
1answer
150 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
1
vote
4answers
57 views

Find matrix determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...
2
votes
3answers
63 views

Is there any way to check wheter the determinant of a matrix $A$ with $|\text{det }A|=1$ is positive or negative?

Let $A\in\text{GL}(n,\mathbb{R})$ with $|\text{det }A|=1$. Is there any way to check wheter $\text{det }A$ is positive or negative without computing it?
0
votes
0answers
20 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
0
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0answers
19 views

Representative value of non-square matrix

First of all, I apologise if this question is inappropriate, I wish I could be more specific - but due to the nature of it, as I am actually asking for a suggestion of some technique, that's hard to ...
12
votes
3answers
191 views

determinant inequality $\det(A^2+AB+B^2)\geq\det(AB-BA)$

$A,B$ are two $2\times 2$ real matrices, then $$\det(A^2+AB+B^2)\geq\det(AB-BA)$$ The inequality is equivalent to the following problem: Let $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
0
votes
2answers
43 views

Odd-dimensional skew-symmetric matrix is singular, even in a field of characteristic 2

I'm familiar with the usual proof $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A)$ which only works in fields of characteristic not equal to 2. To get a proof that works in characteristic 2 I can ...
0
votes
1answer
21 views

Determinant of a matrix over a field K

Let $A$ be an $n \times n$ matrix over a field $K$. Do the properties of the determinant of a real matrix hold for the matrix $A$? If not, in which fields do the properties of the determinant of a ...
0
votes
2answers
27 views

finding determinant for matrix using upper triangle method!

so Here an example for matrix that I'm trying to evaluate its determinant! | 1 3 2 1| | 0 1 4 -4| | 2 5 -2 9| | 3 7 0 1| when applying first row operation i get | 1 3 2 1| | 0 1 ...
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1answer
21 views

meanings determinants of matrices in finite field

Let's $\Bbb{Z}_q$ is finite field. ($q$ is prime number). Lets $A_1$ – set of matrices $n\times n$, such that $\det(M) = 1$, for any matrix $M \in A_1, A_2$ – set of matrices $n\times n$, such that ...
3
votes
2answers
52 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
3
votes
0answers
24 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
7
votes
4answers
253 views

Determinant of a Special Symmetric Matrix

If $A$ is a symmatric matrix of odd order with integer entries and the diagonal entries $0$ then $A$ has determinant value even. I can prove the result if I can show that the eigenvalues of $A$ are ...
2
votes
4answers
101 views

How to prove the inequality $\det (AA^T) \ge 0$?

How to proof for any matrix $A \in \Bbb R^{n \mathbf x n}$, that the next inequality $\det(AA^T) \ge 0$ is true?
3
votes
3answers
63 views

Determining the values of $\lambda$ for which the matrix is invertible

I'm working on a homework problem and am a little stuck. The question is: Determine the values of $\lambda$ for which the matrix $$\begin{pmatrix} \lambda &-1&0\\ -1&\lambda&-1\\ ...