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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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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16 views

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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2answers
212 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
13 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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30 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
24 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 ...
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1answer
59 views

Expressing the determinant in terms of the trace of a matrix and the trace of its square

How can I prove that $$\det(A) = \frac{ 1 }{ 2 } \begin{vmatrix}\operatorname{tr}(A) & 1 \\ \operatorname{tr}(A^{2}) & \operatorname{tr}(A)\end{vmatrix}$$ where vertical bars mean the ...
2
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2answers
60 views

Determinant of the sum of an identity matrix and a rank-two-symmetric matrix

Suppose $I$ is an $n \times n$ identity matrix, and $S$ is the $n \times n$ symmetric matrix with rank equals two. I was reading something saying that: $$\det(I-S)=(1-\lambda_1)(1-\lambda_2)$$ where ...
3
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0answers
55 views

Determinants, how do they emerge and why? [duplicate]

This is a very simple and basic question, what's the simplest proof that the determinant is the same no matter what basis you choose? Also, I've been wondering what exactly is the determinant, what ...
2
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1answer
44 views

Determinant in $\mathbb Z_{5}$

I need to find $$ \det\left[ \begin{array}{cc} 2 & 4 & 0 \\ 1 & 1 & 3 \\ 3 & 2 & 1 \end{array} \right] $$ over $\mathbb Z_{5}$ What I did: $$2\det\left[ ...
1
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1answer
35 views

Finding determinant of a 3x3 matrix

Assuming y is a nonzero real number, I need to find the determinant of this matrix: $$ \left[ \begin{array}{cc} 1 & y & y^2 \\ y & y^2 & y^3 \\ y^2 & y^3 & y^4 ...
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9answers
286 views
+100

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
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1answer
24 views

3 x 3 linear system organization

How to organize this 3x3 linear system in order to solve it with determinants afterwards.
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3answers
38 views

$3\times3$ linear system organization

How to organize the system below? Especially the 2nd row of the system. $$\left\{\begin{eqnarray} 4x-3y+2z+4&=&0\\ x-\frac y3+\frac z2&=&-\frac16\\ 5x+2z&=&3y-3\\ ...
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2answers
85 views

A special case: determinant of a $n\times n$ matrix

I would like to solve for the determinant of a $n\times n$ matrix $V$ defined as: $$ V_{i,j}= \begin{cases} v_{i}+v_{j} & \text{if} & i \neq j \\[2mm] (2-\beta_{i}) v_{i} & \text{if} ...
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1answer
21 views

Calculate the Determinant of a NXN matrix

Is there any elegant way to calculate the determinant of the N X N symmetric matrix M, where the $(i,j)$ term is defined by: $$M_{ij}=m_i+m_j$$ with $0\le m_i, m_j \le1$ The solution will be in ...
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2answers
26 views

Is this Determinant and Trace identity equivalent to Unitary matrix?

Thanks for any help in advance. I have this equality for a 2x2 invertible complex matrix: $$\text{Tr}(AA^*)=2|\text{det}(A)|^2$$ where $*$ is complex conjugate transposition. Is this equality ...
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0answers
37 views

Proof of Minkowski determinant inequality

I wonder where can I find the proof for the Minkowski determinant inequality? ( i.e., given two positive definite n x n symmetric matricies A and B, $det(A+B)^{1/n}\ge det(A)^{1/n}+det(B)^{1/n}$ ) ...
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4answers
350 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
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1answer
65 views

If $I + A + \cdots + A^{n-1} = O$, $A$ a square integer matrix, $n$ odd, for what $k$ does $\det(\sum_{i = k}^{n-1} A^i) = \pm 1$?

This question is, in a sense, homework. I'm taking a problem-solving seminar which uses questions like these, the first question on the 2010 Virginia Tech Regional Math Competition, as fodder. The ...
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1answer
127 views

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
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1answer
75 views

If $A^n = I$, $n$ odd, $A$ a square integer matrix, does $A = I$?

Edit: Crap, even my hypothesis was wrong. If you put $A = \left[ \begin{array}{cc} 1&-1\\3&-2 \end{array} \right]$, then $A^3 = I$ but no eigenvalue is $1$. (What's true is that all ...
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0answers
17 views

Meaning of the determinant of a derivative

I was doing quantum field theory homework and cannot find the meaning of the expression under 'thereom' 3.5 in this document: ...
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0answers
19 views

Determinant of Cauchy matrix

Today I came to know about Cauchy sequence but in wikipedea no proof for the determinant was given. Can anyone help me to understand on this regard? Thanks in advance
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1answer
101 views

Can we determine the determinant?

Could someone prove that this determinant is not zero? $$\left| \begin{array}{cccc} 1^n & 2^n & \cdots & (n+1)^n \\ 2^n & 3^n & \cdots & (n+2)^n \\ ...
5
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2answers
81 views

Determinant of $2\times 2$ Block Matrix

I would like to know the proof for: The determinant of the block matrix\begin{pmatrix} A & B\\ C& D\end{pmatrix} equals $(D-1) \det(A) + \det(A-BC) = (D+1) \det(A) - \det(A+BC),$ when $A$ is a ...
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1answer
30 views

Perturbation of Determinant

Suppose we have a linear equation with parameter $0 <\lambda <1$ as $\left(\begin{array}{ccc} 3-\lambda & -1 & -1\\ -1 & 1-\lambda & 0\\ -1 & 0 & 1-\lambda ...
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0answers
39 views

Determinant - derivation of the general formula and its history [duplicate]

I know the formula for calculating matrix determinant. What's I'm wondering is where did that general formula come from? And why determinants are so important? Obviously they are useful in finding ...
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1answer
22 views

Finding that values k that make this matrix invertible without using the determinant

The matrix in question is A = [(1,1,1),(1,2,k),(1,4,k^2)]. I know that I can row reduce the matrix to rref, which should in theory leave me with some k values in the matrix from which I can see what ...
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1answer
32 views

How to solve matrix eigenvalue equation which has a summation.

General problem: If I have some $n \times n$ matrices $\mathsf{M}^\tau$, and column vectors (with $n$ rows) $X^\tau$ is there some mathematical tricks I can do to solve the eigenvalue equation $ ...
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3answers
49 views

Determinant question $\det(A^{-1/2}) = \det(A)^{-1/2}$

Can someone show me how: $\det(A^{-1/2}) = \det(A)^{-1/2}$ where we assume that $A$ is invertible. thanks
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0answers
27 views

Complex Matrix Determinant Constraints

I am currently a bit stuck on a problem and I would like to get some input to get me going again. I need to solve an optimization problem involving a complex matrix $L$ which depends on the ...
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0answers
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$ Det(cA^{-1})=c^n \frac{1}{det(A)} $

$ Det(cA^{-1})=c^n \frac{1}{det(A)} $ also $ Det((cA)^{-1})=c^n \frac{1}{det(A)} $ Is any of those true?
3
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0answers
35 views

Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
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2answers
48 views

Determinant Question (Proof)

Let $C$ and $D$ be $n \times n$ matrices where n is odd such that $CD = -DC$. Show that either $C$ or $D$ has no inverse. I have no idea how to go about doing this problem. Any help would be ...
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1answer
33 views

Determinant and generalized eigenvalues

Let A, B be two symmetric positive-definite matrices. Let $\lambda_i$ be the generalized eigenvalues of the pencil (A,B). Can we write function $\log\frac{|A|}{|B|}$ (where $|\cdot|$ stands for ...
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1answer
36 views

Matrix: determinant & Diagonal

There is a question that comes up in my mind after I watched Prof. Gilbert Strang's lectures. He was saying: For any matrix $A$, Since $A = LU$, $\det(A) = \det(LU)$ and $\det(L) = 1$, hence $\det(A) ...
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4answers
177 views

Proving determinant using properties of determinants

$$\begin{vmatrix} 1 & 1 & 1\\ a & b & c\\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$$ we have to solve this by using the properties of determinants without ...
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3answers
121 views

Proving determinants using properties of determinants

$$\begin{vmatrix} 1 & a^2+bc & a^3\\ 1 & b^2+ca & b^3\\ 1 & c^2+ab & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a^2+b^2+c^2)$$ we have to solve this by using the properties of ...
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1answer
51 views
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1answer
112 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
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3answers
75 views

Supose $A$ is a 4x4 matrix such that $det(A)=\frac{1}{64}$

Supose A is a 4x4 matrix such that $det(A)=\frac{1}{64}$ then $det(4A^{-1})^T$ I created a 2x2 matrix $B$ and transposed it both had the same deternminant I then found $det(B)$ and $det(B^{-1})$ ...
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1answer
46 views

Linearizing a nonlinear system of ODE about an equilibrium

Since the method below is probably correct, and correctness is potentially irrelevant to my ability to do what I want to learn. Assume below is correct. ...
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2answers
54 views

Matrix with entries from $1$ to $16$, each occuring once, and determinant $40800$

In OEIS, it is claimed, that the largest possible determinant of a $4\ x \ 4$-matrix with the entries from $1$ to $16$, each occuring once, is $40800$. Unfortunately, the article does not mention a ...
3
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2answers
57 views

What is the determinant value of $J-I$ if $I$ is identity matrix and $J=(1)_{101\times 101}$? [duplicate]

Let $J$ be a matrix of order $101\times 101$ which each entry is 1 and suppose $I_{101}$ is identity matrix of order $101\times 101$. The question is : what should be the determinant value of $J-I$ ? ...
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3answers
52 views

Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
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3answers
167 views

Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...
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2answers
60 views

Determinant of identity minus adjacency matrix

Let $M$ be the adjacency matrix of a directed graph $G$. Is there any known relation between $\det(\textrm{id}-M)$ and the cycles of $G$? It is easy to see that if $G$ is acyclic then this ...
12
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0answers
118 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
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2answers
243 views

Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...