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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3
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1answer
22 views

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

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 $J-I$ ? ...
2
votes
2answers
26 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 ...
11
votes
3answers
152 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 ...
0
votes
2answers
50 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
votes
0answers
92 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 ...
3
votes
2answers
143 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 $n\times n$ ...
6
votes
4answers
368 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 ...
1
vote
0answers
24 views

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 ...
3
votes
1answer
68 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 ...
1
vote
1answer
18 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 ...
1
vote
2answers
223 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 ...
1
vote
0answers
54 views

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 ...
5
votes
1answer
70 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 ...
1
vote
1answer
36 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*} ...
0
votes
0answers
53 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}= ...
0
votes
2answers
40 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 ...
2
votes
2answers
66 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 ...
2
votes
2answers
73 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 ...
0
votes
3answers
77 views

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 $\det(A) \det(B)=1*1=1$ ...
2
votes
0answers
42 views

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 ...
1
vote
0answers
17 views

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} ...
0
votes
1answer
64 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 & ...
4
votes
1answer
34 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 ...
11
votes
1answer
276 views

How 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\\ ...
2
votes
0answers
50 views

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 & ...
2
votes
1answer
36 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 ...
1
vote
0answers
39 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 ...
6
votes
0answers
65 views

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 \\ ...
2
votes
1answer
65 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\\ ...
3
votes
2answers
74 views

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 ...
1
vote
2answers
31 views

The determinant of adjugate matrix

Why does $\det(\text{adj}(A)) = 0$ if $\det(A) = 0$? (without using the formula $\det(\text{adj}(A)) = \det(A)^{n-1}.)$
5
votes
1answer
77 views

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 ...
1
vote
1answer
77 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
votes
1answer
66 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 ...
4
votes
0answers
65 views

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 ...
1
vote
1answer
102 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 ...
2
votes
1answer
36 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 ...
1
vote
1answer
34 views

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 ...
1
vote
1answer
48 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. ...
2
votes
2answers
118 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
votes
1answer
57 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 ...
1
vote
1answer
36 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 ...
3
votes
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 ...
1
vote
1answer
37 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 ...
2
votes
0answers
42 views

Probability that a random integer matrix is singular

Let A be a nxn-matrix with integers in the range $u..v$ , where $u<v$ are arbitary integers. Is there a formula, or at least, a good estimate, for the probability that the matrix is singular ? ...
5
votes
0answers
101 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
1
vote
1answer
27 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
0
votes
3answers
91 views

Prove (or disprove) property of determinant: $\;\det(qA) = q^{n} \det(A).$ [duplicate]

Let $A$ be a square matrix. Prove (or disprove) the following: $$\det(qA) = q^{n} \det(A).$$ I tried disproving it with counterexamples but I could not find one. Is there a counterexample I'm ...
1
vote
2answers
77 views

How to find the determinant of this matrix

I'd like to find the determinant of following matrix $$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & ...
4
votes
1answer
59 views

Determinant of sum of matrix with special singular matrix

What is the determinant of the sum of two matrices when one of them is all zeros except for a single column of 1's. I.e. \begin{equation} Det \left[G + S\right] \end{equation} Where \begin{equation} S ...