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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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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164 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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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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55 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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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 ...
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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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54 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
183 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
68 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
152 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
679 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 $n\times n$ ...
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397 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
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Is finding a matrix out of some set with a given determinant a hard problem?

Given $n\ge 2\ \ ,\ u,v,k\ $ integers. Decision problem : Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist? In which complexity class is this problem ? Is it ...
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1answer
73 views

Is det(A) maximal, if det(A+E) is maximal?

Let A be a binary matrix of size n x n and E be the matrix of the same size with all entries $1$. Proof or disproof : If det(A+E) has the maximal possible value, then det(A) also has the maximal ...
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1answer
25 views

Verification Matrices & Linear Equations Part 2

...Continued Question 3 A - True because if it equals 4 then there will be infinite solutions B - True because any gradient except for one that is equal (4) will intersect giving a unique ...
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2answers
250 views

Matrix Equation- solution

Sir, We have given $A= \begin{bmatrix}q_1 & q_2&q_3 \\ q_4 & q_5&q_6\\ q_7 & q_8&q_9 \end{bmatrix} \tag 1$. A is a matrix with determinant 1,orthogonal , invertible and ...
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Teaching determinants

I am writing a first handout on determinants. The intended audience is confident with basic matrix algebra and the basic definitions of vector space theory. I just wondered if someone would comment on ...
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138 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
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1answer
40 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
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57 views

show that the determinants are equal

Prove that the determinants are equal $$ \begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \\ \end{vmatrix}= ...
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85 views

Determinant Formula for Tri-Diagonal Matrix

for an assignment in numerical analysis, I need to find the eigenvalues of a matrix with values only in the diagonal, upper diagonal and lower diagonal. I guess there is an easy formula for this sort ...
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2answers
72 views

Generalization of a formula for 2x2-matrices

It is well known that $$|det(v_1,...,v_n)|\le ||v_1||_2...||v_n||_2$$ with equality if and only if the vectors are pairwise orthogonal. For n = 2, the following formula holds : $$det(\pmatrix ...
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142 views

nth derivative of determinant wrt matrix

I'm working on an expression for the nth derivative of a (symmetric) matrix, i.e. \begin{equation}\frac{\partial^{n} \det(A)}{\partial A^{n}}\end{equation} Starting with \begin{equation}\frac{\partial ...
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Intutive meaning of $\det(AB)=\det(A) \det(B)$.

If we take determinant as volume of unit cube let say A than $\det(A)=1$ as its volume is 1. Now let take another unit cube B and if we put both cubes side by side than then $\det(A) \det(B)=1*1=1$ ...
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Least number not being the determinant of a set of matrices

Let n > 1 be a natural number and u < v integers. How can I determine the least natural number not being the determinant of some n x n - matrix with integers in the range u..v without calculating ...
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Further on determinants and finding the second partial derivative

Below is the question: $$\begin{cases} v+log\left|u\right|=xy \\ u+log\left|v\right|=x-y \end{cases}\implies \begin{cases} ...
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91 views

Show without expanding that the two determinants are equal

$$ Let\ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
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1answer
41 views

How do they go from implicit partial differentiation in this problem to solving with a determinant?

In this book I'm studying I've come across a problem where the author solves a partial differentiation problem using determinants. I'm somewhat familiar with them, but I don't see how they derive the ...
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1answer
360 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
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Strange phenomena in determinants of matrix of determinants.

In my research, my computations are giving rise to the following strange phenomena: Let $$D=\begin{bmatrix}x_1^p & x_2^p & x_{3}^p\\ x_{1}^q & x_{2}^q & x_{3}^q\\ x_{1}^r & ...
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1answer
47 views

Gauss Seidel Method - How do I avoid calculating $L^{-1}$?

I'm trying to write a matlab code that gets a diagonal dominant matrix $A$, vector $b$, and finds an approximate solution $x$ to $Ax=b$ using Gauss-Seidel Method. I understand the theory. Suppose ...
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63 views

Minimum absolute determinant of a regular latin square matrix

It is easy to show that a latin square of size n x n has a determinant, which is a multiple of $\large \frac{n^2(n+1)}{2}$, if n is odd and $\large \frac{n^2(n+1)}{4}$, if n is even. This is a lower ...
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Maximum determinant of latin squares

I strongly conjecture that the maximum absolute determinant of a latin square can be attained by a circulant matrix. For example, $\pmatrix {5&4&2&3&1 \\ 1&5&4&2&3 \\ ...
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1answer
70 views

sign determinant $2\times 2$

I have been reading internet and tried to understand the explanation of the sign of a determinant of a $2\times 2$ matrix. if I have a matrix \begin{array}{cc} a & b \\ c & d\\ ...
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2answers
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Matrix notation why is column 3= column 1?

let $A =$\begin{bmatrix}a_{11} & a_{21} & a_{11}\\a_{12} & a_{22} & a_{12}\\a_{13} & a_{23} & a_{13}\end{bmatrix} where $a_{ij}\in\Bbb R$ for each $1\le i , j\le 3$ which of ...
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The determinant of adjugate matrix

Why does $\det(\text{adj}(A)) = 0$ if $\det(A) = 0$? (without using the formula $\det(\text{adj}(A)) = \det(A)^{n-1}.)$
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Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
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1answer
85 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
3
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1answer
70 views

Surprising necessary condition for a “shift-invariant” determinant

Let $A$ be a $4\ x\ 4$ binary matrix and $Z=\pmatrix {s&s&s&s \\ s&s&s&s \\s&s&s&s \\s&s&s&s}$ Then $\det(A+Z)=\det(A)=1\ $ (independent of s, so ...
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Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
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1answer
109 views

Simple proof that a $3\times 3$-matrix with entries $s$ or $s+1$ cannot have determinant $\pm 1$, if $s>1$.

Let $s>1$ and $A$ be a $3\times 3$ matrix with entries $s$ or $s+1$. Then $\det(A)\ne \pm 1$. The determinant has the form $as+b$ with integers $a$,$b$ and it has to be proven that $a>0$ if ...
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1answer
40 views

Determinant of a matrix shifted by m

Let $A$ be an $n\times n$ matrix and $Z$ be the $n\times n$ matrix, whose entries are all $m$. Let $S$ be the sum of all the adjoints of $A$. Then my conjecture is $\det(A+Z)=\det(A)+Sm$ , in ...
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Relation on the determinant of a matrix and the product of its diagonal entries?

Let $A$ be a $3\times 3$ symmetric matrix, with three real eigenvalues $\lambda_1,\lambda_2,\lambda_3$, and diagonal entries $a_1,a_2,a_3$, is it true that \begin{equation*} \det ...
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1answer
65 views

Properties of Determinant of matrix sum/multiplication

!Hey there :) I am currently working on a topic in control engineering and I'm currently looking for some way to relate determinants of matrix combinations to the determinant of the elements. ...
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2answers
273 views

Prove that if the sum of each row of A equals s, then s is an eigenvalue of A. [duplicate]

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: ...
2
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1answer
77 views

Possibilities of calculate the determinant of an $168\times168$ matrix

Sincerely I've zero knowledge of this kind of math, but I've recently come to work in a friend's project to calculate a matrix of this size. This friend of mine has tried to do this with excel, with ...
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1answer
42 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
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1answer
49 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
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45 views

Evaluation of a Hankel-like determinant

I consider the following determinant (Hankel-like?) $$ [f_1,f_2,...,f_n]:=\begin{vmatrix} f_1 & f_2 & \cdots & f_{n-1} & f_n\\ n-1 & f_1 & \cdots & f_{n-2}& f_{n-1}\\ 0 ...