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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.

0
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2answers
48 views

How many solutions $Ax=0$ has

Problem How many solutions equation $Ax=0$ has when $A$ is defined as: $$ A = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 4 & 0 \\ -4 & 2 & 1 \end{bmatrix} $$ Attemtp to solve ...
1
vote
3answers
37 views

Let $A,B,C,D \in \mathbb{R}^{n×n}$. Show that if $A, C, B−AC^{−1}D,$ and $D−CA^{−1}B$ are nonsingular then that the following matrix has the matrix:

Let $A,B,C,D \in \mathbb{R}^{n×n}$. Show that if $A, C, B−AC^{−1}D,$ and $D−CA^{−1}B$ are nonsingular then $\left[ \begin{smallmatrix} A&B\\ C&D \end{smallmatrix} \right]^{-1} = \left[ \begin{...
1
vote
2answers
20 views

Polynomial and determinant relationship?

It can be proven that If $$P(x)=ax^3+bx^2+cx+d$$ has two roots being opposite of each other i.e. $\alpha$ and $-\alpha$, then $$ad=bc$$ This instantly made me think of a zero determinant of the ...
8
votes
2answers
123 views

is $\det(A^2 + I)$ always non negative?

Obviously $\det(A^2)$ is (casework), but is the above matrix non-negative? $\det(A)\det(A) \geq 0$ as $\det(A) > 0$ or $\det(A) < 0$ yields positive when squared. However, I am not sure that ...
11
votes
2answers
1k views

4x4 Determinant [duplicate]

$A =\begin{vmatrix}a^2 & b^2 & c^2 & d^2\\ (a-1)^2 & (b-1)^2 & (c-1)^2 & (d-1)^2 \\ (a-2)^2 & (b-2)^2 & (c-2)^2 & (d-2)^2\\ (a-3)^2 & (b-3)^2 & (c-3)^2 &...
1
vote
1answer
37 views

Is there any easy way to find the determinant of a 4x4 matrix?

I have a 4x4 covariance matrix and want to find the eigenvalues. I know part of the process is to find the determinant: $$\tiny{\begin{align}\begin{vmatrix} 3.33−\lambda & −1.00 & 3.33 & ...
2
votes
1answer
42 views

Does this property of determinants generalize?

Consider the following $2\times 2$ matrix $$ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}. $$ Let $\Delta = a_{11}a_{22}-a_{21}a_{12}$ be its determinant. Then $A$ ...
0
votes
1answer
21 views

Positive/Negative Definite/Semidefinite Test Generality

A test to determine whether a matrix is positive definite, negative definite, positive semidefinite, negative semidefinite, or none of the above, is to calculate the determinant of every cascading ...
-2
votes
0answers
51 views

To find the number of spanning trees of a graph given its adjacency matrix

A graph $G$ has the following adjacency matrix: $$ \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 &...
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2answers
42 views

Special property of matrix $A\in\mathbb{R}^{n,n}$ and it's determinant

How to prove for matrix $A\in\mathbb{R}^{n,n}$ that if in places where $x$ columns crosses with $y$ rows are placed $0$, then we can be sure, that $$ \det A = 0\qquad \qquad \text{if }\quad x+y > n ...
0
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2answers
39 views

Determinant of matrix $A\in\mathbb{R}^{n,n}$ [closed]

How to prove that for matrix $A\in\mathbb{R}^{n,n}$ we have $$\det A = \det \begin{vmatrix} x & x& x& ...&x&x\\ 1-x& 1&1 & ...&1& 1\\ 0& 1-x& 1&...
1
vote
2answers
32 views

When does $AB$ have linearly independent columns, if $A$ and $B$ are non-square matrices?

If $A$ is $m \times n$ ($m<n$), and its rows are independent $B$ is $n \times p$ ($p<n$), and its columns are independent We also know $m\ge n$. does $AB$ have linearly independent columns? ...
4
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1answer
153 views
+50

Integral of a determinant over a unit simplex

Recalling the definition of the unit simplex $$ \Delta^{(n)}=\lbrace (x_1,\dots,x_n)\in \mathbb{R}_+^{n} \; , \sum_{i=1}^n x_i=1 \rbrace,$$ I would like to calculate this integral for all integers ...
3
votes
3answers
132 views

Finding determinant of $3\times3$ matrix

$$A = \left(\begin{matrix} \lambda - 1 & -1 & -1 \\ 1 & \lambda - 3 & 1 \\ -3 & 1 & \lambda + 1 \\ \end{matrix}\right)$$ $$\det A = \begin{vmatrix}A\end{vmatrix} = (\lambda - ...
1
vote
3answers
55 views

Compute this determinant only using determinant properties

Compute $$\begin{vmatrix} 1 & x & x^2 & x^3 \\ 1 & y & y^2 & y^3 \\ 1 & z & z^2 & z^3 \\ 1 & x+y+z & xy+yz+zx & xyz \end{vmatrix}$$,where $x,y,z \in \...
1
vote
1answer
25 views

Sign rule for finding the adjugate of a 3x3 matrix?

So i have this matrix A= $$ \begin{pmatrix} 1 & 3 & 0 \\ -2 & -5 & 2 \\ 1 & 4 & 3 \\ \end{pmatrix} $$ And i want to find the inverse of it. Following all ...
3
votes
3answers
60 views

$\det(A^2+A-I_2)+\det(A^2+I_2) = 5$

Let $A \in M_{2\times 2}(\mathbb{C})$ and $\det(A)=1\DeclareMathOperator{\tr}{tr}$ Prove that $\det(A^2+A-I_2)+\det(A^2+I_2) = 5$ using Cayley-Hamilton Theorem $A^2-\tr(A)A+\det(A)I_2=0$ $\det\big(\...
0
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0answers
38 views

Show $ f'(0)=\det(A)\operatorname{tr}(A^{-1}B)$ [closed]

Let $A,B \in \mathbb{M}_{n\times n}(\mathbb{R})$ be square matrices with real coefficients, and consider the function $$ f(t)=\det(A+tB) $$ Show that $f$ is a polynomial in $t$, and that for ...
2
votes
3answers
49 views

Computing an almost Vandermonde matrix

I have this determinant which looks like a Vandermonde matrix $$D=\begin{vmatrix}1& a_1 & \cdots & a_1^{n-2}& a_1^n\\ 1& a_2 & \cdots & a_2^{n-2}& a_2^n\\ \vdots &\...
6
votes
2answers
1k views

A pattern in determinants of Fibonacci numbers?

Let $F_n$ denote the $n$th Fibonacci number, adopting the convention $F_1=1$, $F_2=1$ and so on. Consider the $n\times n$ matrix defined by $$\mathbf M_n:=\begin{bmatrix}F_1&F_2&\dots&F_n\...
0
votes
5answers
50 views

$A^TJA = J$ $ \rightarrow \det(A) = 1$ [on hold]

Why is this true? I cant seem to understand why the $\det(A) = 1$ for this to hold?
-1
votes
3answers
58 views

Proving that $\text{tr}^2(A) - 4\det(A) \geq 0$

I'm trying to figure out how to prove that when given a diagonal matrix \begin{align} A= \begin{bmatrix} \lambda & 0\\ 0 & \mu \end{bmatrix}, \end{align} with a positive determinant $\det(A) &...
2
votes
1answer
33 views

Determinants and uniqueness of a solution

I have a question here which says: "If det$(A)\neq0,$then $Ax=x$ has a unique solution." The $x$ on the right side isn't a typo. It's not supposed to be $b.$ I am supposed to show that this is true ...
2
votes
1answer
104 views

What is the relationship between the determinant and the derivative of a linear map?

I know determinants tell you the oriented volume of the parallelepiped after the linear transformation, but if you define the derivative as I do below then it seems equivalent, at least in terms of ...
0
votes
2answers
57 views

Determinant of the $n\times n$ matrix whose $jk$-th entry is $1/\min\{j,k\}$

So on my exam I got a True/False question that asked the following: For any $n\in\mathbb N$, compute the determinant of the matrix \begin{bmatrix} 1&1&1&\cdots&1 \\1&1/2&...
0
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0answers
41 views

Matrix determinant inequality

Knowing that $A$ and $B$ are $2$ matrices of $2\times2$ with elements on $\mathbb{R}$, we have to prove that $\det(A^2+B^2)\ge\det(AB-BA)$ I tried calculating it directly and got nothing. Any ideas?
2
votes
0answers
61 views

Is there a generalization of Pfaffians?

For an skew-symmetric matrix $A$ (meaning $A^T=-A$), the Pfaffian is defined by the equation $(\text{Pf}\,A)^2=\det A$. It is my understanding that this is defined for anti-symmetric matrices because ...
3
votes
2answers
59 views

Find the determinant of $N \times N$ matrix

I have the following $N \times N$ matrix. \begin{vmatrix} 0 & 1 & 1 & \ldots & 1 \\ 1 & a_1 & 0 & \ldots & 0 \\ 1 & 0 & a_2 & \ldots & 0 \\ \vdots &...
1
vote
2answers
25 views

Finding a function with arbitrary Jacobian determinant everywhere

If we have a function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ and $ \forall x, g(x) > 0$, can we always find a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ s.t. $\forall x, |\det \frac{\...
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2answers
28 views

homogeneous only trivial or infinite

This question has three parts. There are similar questions on stack exchange, but if you read all of the questions, then you'll see that this is not a duplicate [at least not one that I could find.] ...
0
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0answers
30 views

Interpreting a statement regarding Determinants.

Let A be any matrix. Let S be any set of rows of A. What does the statement "each minor determinant in S which has as many rows as S" mean? Doesn't minor determinant of S has necessarily lesser rows ...
0
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0answers
83 views

Computing determinant

Let $$ A= \begin{bmatrix} x_1^{4} & x_2^{4} & x_3^{4} \\ x_1^{2} & x_2^{2} & x_3^{2} \\ 1 &1 &1 \end{bmatrix} .$$ I want to show that the $$\det A \times \frac{1}{(x_1-...
2
votes
1answer
35 views

Extracting the phase of a determinant

Is it possible to extract the phase of a determinant without computing the full determinant? More explictly, given a complex matrix $U$, the determinant can be written in the form \begin{equation} \...
2
votes
3answers
68 views

Determinant of $(I+uv^*)$

Let $u,v \in R^n$, How can I find $det(I+uv^*)$? This problem was given as preparatory for final exam and I dont know how to approach it. I dont see some nice ways to express this determinant. This is ...
0
votes
0answers
23 views

Why is the volume of a parallelepiped linear in each row in the matrix representation?

In this question, the determinant of a matrix is explained to be a measure of the volume of a parallelepiped formed by using the columns in a matrix as vectors. It is also noted that the determinant ...
1
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0answers
35 views

Orientation of 4 + 1 lines in $\mathbb{R}^3$.

I'm working on a 3D algorithm that at some point establishes orientation of two lines - the same way one would do using the triple product. The way those lines are described, however, makes the ...
0
votes
1answer
31 views

The tensor $\epsilon_{ijk}$ is related to determinants?

My textbook says the following in an appendix on tensor notation: The tensor $\epsilon_{rst} = \begin{cases} 0 \qquad & \text{unless $r, s,$ and $t$ are distinct} \\ +1 \qquad & \text{if $...
3
votes
1answer
47 views

Computing Determinant of a Matrix $\textrm{det}(A^{101} - A)$

Let $A$ be a $n$-by-$n$ matrix with real entries such that $A^{-1} = 3A$. What is the determinant of $A^{101} - A$ i.e. $\textrm {det}(A^{101} - A)$? My attempt: $$A^{-1} = 3A \Rightarrow 3A^{2} = I ...
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6answers
70 views

Finding the determinant of {{1,2,3}, {4,5,6}, {7,8,9}} by inspection.

$$ \det \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} = 0 $$ Is it possible to find the determinant of the matrix $$ \begin{bmatrix} 1 & 2 ...
1
vote
2answers
35 views

Existence of a interpolating polynomial of degree less than or equal to 3, given only 3 support points and 1 derivative value

I'm trying to show the existence and uniqueness of an interpolating polynomial $p$ of degree less than or equal to 3 that interpolates a differentiable function $f$ ( i.e. $f(x_i) = p(x_i)$ for $i = 0,...
2
votes
1answer
72 views

matrices and determinant

Let $A\in \mathcal{M}_{n,p}(\mathbb{R})$ and $G={}^{t\!}AA$ We assume that $\operatorname{rk}(A)=p$ To show that $\det(G)>0$ the argument provided is $\det G=(\det A)^2>0$, unfortunately I ...
0
votes
1answer
67 views

Sylvester's determinant identity proof problems

So I was stuck with the famous Sylvester's determinant identity and don't know how to correctly interpret the difference between the sizes of identity matrices $I_m$ and $I_n$. The line that I dont' ...
0
votes
1answer
23 views

Strange row reduction method

This is a determinant problem with the solution. But I don't understand what sort of row reduction is being used or why. I can see the first term of each row is being subtracted from the others, but ...
1
vote
0answers
51 views

How to find the value of this determinant?

I'm wondering how to find the value of this determinant. $$\left[ {\begin{array}{*{20}{c}} 0&{{x_1}}&{{x_2}}&{{x_3}}& \ddots &{{x_n}} \\ {{x_1}}&0&{{x_1}}&{{x_2}}&...
0
votes
0answers
32 views

Is there an easy way to find the sign of this determinant without calculating it directly?

There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, the determinant appears ...
4
votes
0answers
30 views

Matrices with positive permutation products

Let $A=(a_{ij})$ be a $n\times n$ real matrix such that $$ \operatorname{sign}(\sigma) \cdot a_{1,\sigma(1)} a_{2,\sigma(2)} \ldots a_{n,\sigma(n)}\ge 0 $$ for all $\sigma\in S_n$. Is there a name ...
1
vote
1answer
37 views

Explanation of Cartesian formula for circumcenter

On Wikipedia there is a Cartesian formula for the circumcenter of a triangle. That is, given points $A$, $B$ and $C$ in $\mathbb{R}^2$, find point $U$ such that $d(A,U)=d(B,U)=d(C,U)$. The formula, as ...
1
vote
0answers
14 views

How do I find the determinant of a n-th row matrix?

How do I solve this type of matrix? I don't know what's the best approach, which row should I deduct, which should I add. Any tips are greatly appreciated. Thanks.
1
vote
1answer
37 views

Solving Vandermonde equation system

Given the following: $$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \...
4
votes
1answer
64 views

Proving that $det(A) \ne 0$ with $A$ satisfying following conditions.

I am given $A \in M_n(\mathbb{R})$ which satisfies the following conditions. $A_{i,i} \gt 0$ for all $1 \le i \le n$ $A_{i,j} \le 0$ for all distinct $1 \le i, j \le n$ $\sum_{j=1}^n A_{i,j} \gt 0$ ...