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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16 views

Check Vector3 points on one line using a Matrix

I know that for 3 Vector2 points (say points a, b, c) the determinant of the following ...
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1answer
24 views

Minimal polynomial of a $4\times4$ matrix [closed]

I just need to see an example of a non-diagonalizable $4\times4$ matrix over $\mathbb{R}$ whose minimal polynomial is the same as its characteristic polynomial. I saw the question elsewhere and ...
1
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1answer
36 views

How do we deduce that it is the zero function?

Theorem If the Wronskian of $x^{(1)}(t), \dots, x^{(n)}(t)$, that are solutions of $x'=Ax$ on an interval, gets zero at some point $t=t_0$ of the interval, then $x^{(1)}, \dots, x^{(n)}$ are linearly ...
2
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1answer
81 views

If $\det{(H_{p}^{\infty})}$ converges to a constant value, estimate the range of $p$.

Introduction: One day I calculated the value of determinant which is like Hilbert matrix $H_{p}^{n} \in \bf{R}^{\it{n \times n}}$using my computer. The determinant is defined below. $$ ...
2
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2answers
31 views

Determinant of complex matrix with almost constant lines

Let $0\neq c\in\mathbb{C}$. Take the matrix $$A_C=\begin{pmatrix} n&c&\dots&c&c \\ c&n&c &\dots & c\\ c &c & n &c &\dots\\ \vdots ...
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1answer
22 views

Transform the determinant to upper triangular form

I have difficulty transforming this determinant $$\begin{vmatrix} x&x & x &\dots & x& x\\ a_1 &x &x &\dots &x& x\\ 0& a_2& x&\dots &x& x\\ ...
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0answers
74 views

$AB - BA$ is invertible and $A^2 + B^2 = AB \implies 3 | n$ [closed]

Given $A,B \in \mathbb M_n (\mathbb R)$ and that $A^2 + B^2 = AB$ $AB - BA$ has inverse Prove that $3 \mid n$.
4
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2answers
567 views

Is there an easy way to compute this determinant

\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & a \\ 0 & 1 & 0 & 0 & a & 0 \\0 & 0 & 1 & a & 0 & a \\0 & 0 & a & 1 & 0 & a \\0 & ...
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0answers
21 views

Having trouble to find the value of the following determinant

I came across the following problem from the book Higher Algebra(by barnard and Child) that says: Prove that $\,\,\begin{vmatrix} bc &bc'+b'c &b'c' \\ ca& ca'+c'a &c'a' \\ ...
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1answer
25 views

Prove a Block Matrix is Positive-Definite Given the elements

Suppose I have a non-negative, symmetric $(n+1)\times(n+1)$ block matrix $$ M = \begin{bmatrix} A & B \\ B^T & 1 \end{bmatrix} $$ where $A$ is an $n\times n$ positive-definite matrix and $B$ ...
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2answers
31 views

Computing the determinant of a matrix which splits up into unequal blocks

Suppose I have a matrix of the following form: $M = \begin{bmatrix} A & \mathbb{v} \\ \mathbb{v}^T & a\\ \end{bmatrix}$ where $A$ is an $n \times n$ matrix, $\mathbb{v}$ is an $n ...
2
votes
1answer
44 views

Determinant of nth order

I want to solve the following determinant: $D_n= \begin{vmatrix} a_n & a_{n-1} & \cdots & a_2 & x\\ a_n & a_{n-1} & \cdots & x & a_1\\ a_n ...
0
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1answer
24 views

Logarithm of complex matrix

For invertible matrix $A$, we have $\log(\det A) = \mathrm{tr}(\log A)$ due to a corollary of Jacobi's formula. What if we had the argument $iA$ instead? Would the above relation still hold? Edit: ...
0
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1answer
21 views

How to prove the value of a “negative” matrix

So I came across the formula that $\det(-A)=(-1)^n \cdot det(A)$, where $n$ is the number of columns/rows of A. I know how you get the formula by Laplace's formula and only described in words somehow, ...
0
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0answers
27 views

Proove that $\det(L_B) = \big(\det(B)\big)^n$, $\det(R_B) = \big(\det(B)\big)^n$. [duplicate]

Let $B \in M_{n×n}(\Bbb F)$. Define the functions $L_B$ and $R_B$ by: $L_{B}(A) = BA$ and $R_{B}(A) = AB$. Prove that $\det(L_B) = \big(\det(B)\big)^n$, $\det(R_B) = \big(\det(B)\big)^n$. $L_B$ and ...
2
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2answers
34 views

Show that determinant is divisible by f(x) [closed]

Let $\alpha$ be a repeated root of the quadratic equation $f(x)=0$ and $A(x),B(x),C(x) $be polynomials of degree 3,4 and 5 respectively.Then show that \begin{vmatrix} A(x) & B(x) & C(x) \\ ...
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1answer
39 views

Theoretical question about rank and invertibility of a block matrix,

Let A and B be real matrices, A is symmetric, and B has at least as many columns as rows. $$ C= \begin{bmatrix} A & B^t \\ B & 0 \\ \end{bmatrix} $$ a) Prove ...
2
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2answers
50 views

Find the inverse and determinant of A=(aI +T),

where is $a\ne 0$, $T$ has rank-one and zero trace. I just verified that a rank-one matrix has at most one non-zero eigenvalue. Now since T is of rank-one and has zero trace, that means all of its ...
6
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1answer
114 views

Determinant of an unknown matrix.

Let $x, y$ be two real variables. If $A$ is any $n\times n$ matrix with all entries in the set $\{x,y\}$ then prove that \begin{equation} \det A = (x-y)^{n-1}(Px + (-1)^{n-1}Qy) \end{equation} where ...
2
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1answer
80 views

Matrices - One problem with two optional solutions, don't know which one is correct

I have the following question: Let $A$ be a $3\times 3$ matrix such that $|\text{adj}(3A)|=3$. Find $|A|$. I solved the question in two different methods, but one method gave a solution which ...
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0answers
23 views

Prerequisites for Blyth's Module theory - an approach to linear algebra

I would like to know from people who have read this book if it can be tackled without prior exposure to linear algebra. The author claims in the introduction that "algebraic prerequisiste is the ...
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1answer
63 views

For what values of $x$ is $\det\begin{bmatrix} x & 5\\ 7 & 10 \end{bmatrix} = 30$?

If I have a matrix of the form $$ A = \begin{bmatrix} 4a & b\\ 4b & a \end{bmatrix}, $$ then $$ \left | A \right | = 4a^{2} - 4b^{2} = 4\left( a^{2} - b^{2} \right) = 4 \left( a + b \right) ...
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4answers
71 views

Prove that the determinant is $0$ by expressing as a product

I need to prove that the determinant $$\begin{vmatrix} my+nz & mq-nr & mb+nc \\ kz-mx & kr-mp & kb-ma \\ nx+ky & np+kq & na+kb \end{vmatrix}=0$$ In my book it is given ...
3
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1answer
19 views

Proof based problem related to non-trivial solution of a linear equation system

If the system of linear equations $$a(y+z)-x=0$$ $$b(z+x)-y=0$$$$c(x+y)-z=0$$ has a non-trivial solution $(a,b,c \neq -1)$,then show that $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}=2$$ ...
2
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0answers
37 views

Looking for a proof of a known theorem involving integral quadratic forms

Let $n$ be a positive integer and let $Q$ be an integral quadratic form in $n$ variables. Let $M$ be the symmetric "two's in" matrix associated with $Q$ so that $Q$ can be expressed as the $1 \times ...
3
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0answers
60 views

Help me to prove the determinant of given matrix.

Suppose, $ M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1}+x & -x-a_2 & 0&0&\cdots &0\\ a_1+x&0 & -x-a_3 ...
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1answer
70 views

Proof of a determinant expansion

This is equivalent to a result in Prasolov's book on linear algebra whose proof is not clear to me. I need help in understanding why the result is true. Let $x_1,x_2,\dots,x_n$ be row vectors in ...
2
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1answer
48 views

The determinant of a certain matrix

How to compute the following determinant? $$\left| \begin{matrix} 1 & x_0 & x_0^2 & \ldots & x_0^n \\ 1 & x_1 & x_1^2 & \ldots & x_1^n \\ \vdots & \vdots & ...
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1answer
23 views

What does it mean for vectors of a matrix to be linearly dependent?

I'm studying matrices and the implications of the determinant being $0$. I've read that if the determinant of a transformation matrix is $0$, then the vectors in the rows or columns are "linearly ...
2
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1answer
29 views

Is every “weakly square” matrix either a $0$ matrix, or a square matrix?

Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly, every square matrix is weakly square, and every zero matrix is weakly square. Question. Are ...
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1answer
79 views

Determinant $n\times n$ problem

$$ D_n = \left| \begin{matrix} n & -1 & -3 & 0 & 0 & \cdots & 0 & 0 & 0 \\ n & 1 & 2 & -3 & 0 & \cdots & 0 & 0 & 0 \\ n ...
2
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1answer
37 views

Determinant problem $n\times n$ with $0$'s and $1$'s

$$D_n=\left\vert\begin{matrix}0&1&0&0&\cdots&0&0\\ 1&0&1&0&\cdots&0&0\\ 0&1&0&1&\cdots&0&0\\ ...
2
votes
3answers
50 views

The Hadamard determinant problem: understanding a proof for the upper bound on matrix determinants

I'm working through a proof on The Hadamard determinant problem which can be found in Proofs from THE BOOK. I don't understand how the transition from real valued matrices $A$ with entries in ...
4
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2answers
53 views

Polynomial of $3\times 3$ matrix with $\operatorname{tr}(A)=\operatorname{tr}(A^2)=0$ and $\det(A)=1$

If $A$ be an invertible matrix $M_3(\mathbb{R})$ and we have $\operatorname{tr}(A)=\operatorname{tr}(A^2)=0$ and $\det(A)=1$, then what is the characteristic polynomial of $A$?
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1answer
29 views

Wronskian determinant and linear independence

I can't understand this fact about the wronskian determinant in differential equations. Consider a homogeneous second order linear differential equation with constant coefficients. Taken two ...
0
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1answer
62 views

Finding inverse of matrix using its trace

Is there a way to find the inverse of a matrix using its trace? Maybe by using "Eigendecomposition"(sometimes called "Spectral Decomposition") of a matrix? If it is given that it has an inverse and ...
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1answer
54 views

Calculating $\det(A)$ using permutations.

So the matrix is $A \in M_{n\times n}(\mathbb R)$ and defined: In row $i$ $1 \le i \le n$ there is $1$ in columns $i+1$ and $i-1$, the rest are $0$'s. Except for: $a_{(1,2)}$=1 $a_{(1,n)}$=1 ...
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0answers
54 views

Jacobian Derivation

I'm not sure whether this question is appropriate to ask here but I was wondering if someone could provide a simple derivation of the Jacobian matrix (or determinant).
0
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1answer
26 views

Projection of a plane on coordinate planes

$$\left|\begin{matrix} x & y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \end{matrix}\right|=0$$ This is the equation ...
2
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2answers
43 views

Inner product space, prove $\det(A) \geq 0$ given a particular matrix $A$

Let $K=\mathbb R$ and let $V$ be a $\mathbb K-$finite dimensional inner product space, $\dim(V)=n$. Consider $v_1,...,v_m \in V$ with $1 \leq m \leq n$. Let $A \in \mathbb K^{m\times m}$ defined as ...
9
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1answer
115 views

What's the geometric meaning of a negative determinant?

Geometrically, the determinant of a matrix is the signed volume of a unit cube after the transformation defined by the matrix is applied. However, I'm have trouble understanding what the "signed" mean ...
2
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3answers
68 views

Finding eigenvalues of an anti-diagonal matrix

Let $A=(a_{ij}) \in M_{n}(\mathbb{R})$ such that $$a_{ij}= \begin{cases} \hfill 1 \hfill & \text{ if $i+j=n+1$} \\ \hfill 0 \hfill & \text{ otherwise} \\ \end{cases} ...
2
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1answer
45 views

How to find determinant of the matrix $N\times N$

How to find determinant of the matrix NxN? $$ \left| \begin{array}{ccccc} a^{n} & (a - 1)^{n} & \cdots & (a-n)^{n} \\ a^{n - 1} & (a - 1)^{n - 1} & \cdots & (a-n)^{n - 1} \\ ...
3
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1answer
61 views

Determinant of a $2n$ square block matrix in which all blocks commute

Problem: Let $A , B , C , D$ be commuting $n$-square matrices. Consider the $2n$-square block matrix $$M=\begin{pmatrix} A & B \\ C & D\end{pmatrix}$$ Prove that $|M|= |A||D| - |B||C|$, ...
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1answer
59 views

Calculate the determinant of the matrices $a_{ij}=\frac{1}{i+j-1}$ and $b_{ij}=\frac{1}{i+j}$?

I would like to know if there is any formula for calculating determinants of the following symmetric matrices: $$ A=[a_{ij}]_{n\times n},\qquad a_{ij}=\frac{1}{i+j-1}, $$ and $$ B=[b_{ij}]_{n\times ...
3
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0answers
21 views

Does the sum of weights in Kirchhoff’s construction equal the Gram determinant?

Background: An electrical network is modeled by a complex. Branch current distributions $\mathbf I\in C_1$ are represented by $1$-chains; branch voltage drop distributions $\mathbf V\in C^1$ are ...
3
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0answers
23 views

Jacobian Determinant vs. Divergence for local expansion

I am interested in image processing (in 3D). I often see two different ways of measuring local expansion or contraction of a deformation: the Jacobian determinant or the divergence (but usually the ...
2
votes
1answer
63 views

Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ [duplicate]

Let $\mathbb{R^{ n×n}}$ be the space of n × n real matrices. Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ is the sum of the determinants ...
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2answers
61 views

Check if a point lies in a circle defined by three other points.

I'm learning Computational Geometry, and need to check whether a point p lies inside a circle defined by a triangle(made by 3 points $a,b,c$, in counterclockwise order). A very convenient method is ...
1
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2answers
39 views

Proving determinant value zero: $\det(M-I)=0$ [duplicate]

If $M$ is a $3\times3$ matrix, where M'M=I and $\det(M)=1$, then prove that $\det(M-I)=0$ By the information given I know that given matrix is orthogonal. How can prove the above determinant zero?