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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Linear systems, eigenvectors

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
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1answer
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A hard exercise on endomorphisms and determinants

The following exercise has been bugging me for some days, could someone help me with it ? Let $E$ be a $\mathbb{C}$-vector space with dimension $n$ and $f\in\mathcal{L}(E)$ ($\mathcal{L}(E)$ denotes ...
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Prove that the determinant of this matrix is non-zero.

Prove that the determinant of this matrix is non-zero for every possible combination of + and - .$$\left[\begin{array}{cc} \pm 1 & \pm 3 & \pm 4 \\ \pm 3 & \pm 2 & \pm 5 \\ \pm 4 ...
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47 views

How to prove the determinant?

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
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Determining matrix in terms of determinants of other matrices.

Determine |a+b e-f| |c+d g-h| in terms of the determinants of |a c| |b d| |a c| |b d| |e g| |e g| |h f| |h f| ...
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30 views

Prove the following determinant without expanding

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
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0answers
22 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...
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1answer
12 views

What is a cartesian equation for 3 space passing through 3 points?

What does cartesian equation for 3 pace look like? and is there any way to describe this equation using determinant?
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1answer
33 views

How do we establish the existence of fundamental matrix of a Markov chain?

Let $X=\{X_n\}$ be a finite state Markov Chain with the state space $S = \{0,1,2,...,N\}$ such that $0$ is the single absorbing state and all the rest states are transient. The following is the ...
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17 views

About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms)

my question is originated from a physical problem. I will try to present the problem as simple as possible, but I fear it will still be long since I'm bad at expressing myself briefly. It starts with ...
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4answers
188 views

Question about determinants

I am working on some practice problems and I just cant see to even begin to understand how to do this question. It starts off by giving some facts such as det= 1 for the following:$$ \begin{matrix} a ...
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1answer
68 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
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22 views

Determinant of specific infinite matrix

What is the limit, as n approaches infinity, of the determinant of an n x n matrix where each cell has the value cos(n * row + column)? My friend and I believe the answer to be 0, but can't figure ...
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1answer
47 views

9 by 9 matrix: finding the determinant?

Can it be done analytically? I have a system I need to solve, but would need to take a determinant of a 9 by 9 matrix. Is it worth the effort, or is there a limit (in rank) above which it's not ...
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+50

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
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1answer
48 views

Expressing determinant as a linear combination of minors of fixed dimension

Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of ...
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2answers
99 views

Give conditions on a,b,c, and d such that A has two, one, and no eigenvalues?

I am given that matrix $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ and I need to find conditions on a,b,c, and d such that A has Two distinct ...
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2answers
29 views

Linear Algebra - Invertible matrices and determinants

Let $A$ be any $n \times n$ invertible matrix, defined over the integer numbers. Let assume that $A^{-1}$ (Inverse of A) is also defined over the integer numbers. Prove that $\det A\in\{-1,+1\}$. ...
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4answers
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Let $A$ be a $3\times3$ matrix. Given $\mathrm{adj}(A)$, find $\det(A)$.

Let $A$ be a $3\times3$ matrix such that $$\mathrm{adj}(A) = \begin{pmatrix}3 & -12 & -1 \\ 0 & 3 & 0 \\ -3 & -12 & 2\end{pmatrix}.$$Find the value of $\det(A)$. I know that ...
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4answers
70 views

Use row reduction to show that the determinant is equal to this variable.

Show determinant of: \begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix} is equal to $(b - a)(c - a)(c - b)$ I'm not sure if you can use squares or square roots hmmm.. ...
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2answers
34 views

Show that a determinant is equal to this variable.

Show that the : determinant of: \begin{pmatrix}0&0&a_{13}\\0&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix} is equal to $-A_{13}A_{22}A_{31}$ I believe the cofactor and ...
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0answers
27 views

Differential equations, derivative of determinant, Euler's formula

Let $b:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth vector field. Let $u(s,x,t):\mathbb{R}^{n+2}\to\mathbb{R}^n$ with $s,t\in\mathbb{R}$ and $x\in\mathbb{R}^n$ satisfy the following differential ...
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4answers
87 views

Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$

Suppose that $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging the first two rows of $A$. Prove that $\det(B) = -\det(A)$. By general $n \times n$ matrix, I mean ...
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Proof of the conjecture that the kernel is of dimension 2

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix ...
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1answer
17 views

Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
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1answer
26 views

Negative determinant

Let $$ A = \begin{bmatrix} -a_{12}-a_{13}-a_{14} & a_{12} & a_{13} & 1\\ a_{21} & -a_{21}-a_{23}-a_{24} & a_{23} & 1\\ a_{31} & a_{32} & -a_{31} - a_{32} - a_{34} & ...
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3answers
41 views

Determinant of linear transformation

Given a linear transformation $T:V\rightarrow V$ on a finite-dimensional vector space $V$, we define its determinant as $\det([T]_{\mathcal{B}})$, where $[T]_{\mathcal{B}}$ is the (square) matrix ...
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1answer
63 views

Determinant of the matrix $\binom{m_i}{j-1}$

Let $m_1,\dots,m_n$ be real numbers $\ge n-1$. How can I find the determinant of the matrix $A$ defined by $(a_{i,j})=\binom{m_i}{j-1}$, for $1\le i\le n$ and $1 \le j \le n$ ? This all looks ...
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3answers
141 views

Determinant of $a_{i,j}=(x_i+y_j)^k$

How can I find the determinant of the matrix $A\in\mathcal{M}_n(\mathbb{R})$ with coefficients $a_{i,j}=(x_i+y_j)^k,k<n$ ? All the $x_u,y_u$ are real numbers. Derivating won't help, and I didn't ...
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1answer
55 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
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1answer
17 views

Determinant of block matrix when $CD^T=DC^T$

When $CD^T=DC^T$ and $D$ is invertible we have: $$\left(\begin{array}{cc} A & B\\ C & D\\\end{array}\right)\times\left(\begin{array}{cc} D^T & 0\\ -C^T & ...
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26 views

Matrix Inverse Question- Singular Matrix issue

I have a given Matrix equation $R(s)^{'}_{3\times 3} = \psi(s)_{3\times 3}R(s)\tag 1$ Conditions R(s) is orthogonal and determinent 1. Can say in the format of rotation matrix $R^{'}(s)$ ...
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1answer
39 views

Do Tensors have a determinant property?

We know that only square $n \times n$ matrices have a determinant property! And it can be defined just like this: $$A=\begin{array} & & & \\ ...
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2answers
115 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
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1answer
42 views

Demonstrate using determinant properties that the determinant of matrix “A” is equal to, 2abc(a+b+c)^3

How can I show, using determinant properties of matrix, that: \begin{equation} \det\begin{pmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & ...
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2answers
56 views

To prove $\det (xy^t)=0$ [duplicate]

Let $x,y$ be arbitrary non-zero column vectors in $\mathbb R^n$ , then how do we prove that $\det (xy^t)=0$ ?
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1answer
59 views

Find the determinant of a symmetric matrix

How can we find the determinant of the following matrix $A$: $\left( \begin{array}{cccccc} x_1y_1 & x_1y_2 & x_1y_3 & \cdots & x_1y_{n-1} & x_1y_n \\ x_1y_2 & x_2y_2 & ...
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45 views

invertible matrix and upper triangular matrix

If we are given a $A$ as $2\times2$ matrix. How to find an invertible matrix $P$ and a upper triangular matrix $U$ such that $A=PUP^{-1}$?
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111 views

Leading principal minors

How many leading principle minors are there for a 4X4 matrix? please explain in detail. I know for a 3X3 matrix.
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1answer
26 views

odd determinants

How many 4 × 4 matrices with entries from {0, 1} have odd determinant? is there a short way of finding the answer to this question or do we have to solve it by hit and trial or using lengthy methods. ...
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1answer
24 views

How do i prove the Leibniz formula of the determinant over a commutative ring?

Let $R$ be a commutative ring. My definition for the determinant over $M_n(R)$ is defined inductively as $\det_{n+1}(A)=\sum_{j=1}^n (-1)^{1+j}A_{1j} \det_n(\tilde{A_{ij}})$. (Here, $(-1)$ denotes ...
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1answer
29 views

calculating determinant

Let M be real vector space of order $2\times3$ matrices with the real entries. Let $T:M\longrightarrow M$ be defined by $T\Bigg( \begin{pmatrix} x_{1} & x_{2} & x_{3} \\ x_{4} ...
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A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
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Indices question in this multilinear algebra question.

Suppose $V$ is finite dimensional with $\dim V = n$ and $f : V \to V$ be linear. Prove that there is a number $d(f)$ such that $$\Omega^n(f)(\omega) = d(f)\omega.$$ Here, $\Omega^n(V)$ denotes the ...
3
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3answers
79 views

Easiest Method to Evaluate $3\times 3$ Determinants

After a lot of practice, I developed a method of evaluating $3\times 3$ determinants which I call the Cross - Left Fish - Right Fish. The method goes like this, for some $3 \times 3$ determinant ...
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Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$ x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t), $$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
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221 views

A determinant problem

If $f(n)=\alpha^n+\beta^n$ and $$A=\left| \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \right|$$ ...
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1answer
17 views

Discriminant of a ternary quadratic form

What is the discriminant of a ternary quadratic form $x^2-y^2+z^2-2xy+4yz-6xz$? The answer says, first make it $a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{13}xz$, and then the discriminant ...
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32 views

Eigenvalues and Determinants of Two Matricies

Suppose $B=[v,e]$ is an $n \times 2$ matrix with $v=[v_1,...,v_n]^T$ and $e=[1,...,1]^T$, and $J_{2\times 2}=[(0,1),(1,0)]$, and so $Rank(BJB^T)=2$. How can we prove that $BJB^T$ and $JB^TB$ have the ...
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1answer
36 views

determinant in terms of quadratic form evaluated at a point

Say $A$ is a $n$ by $n$ positive definite matrix. Let $b$ be a column vector in $\mathbb{R}^n$. Consider the following quantity: $$b^TA^*b$$ where $A^*$ is the cofactor matrix of $A$. A simple ...