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.

learn more… | top users | synonyms

3
votes
0answers
38 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
35 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
62 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
56 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
0answers
68 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
32 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
31 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
43 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
104 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
50 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
34 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
48 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
33 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
39 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 ? ...
4
votes
0answers
65 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
68 views

Determinants Proof

Let A and B be square matrices. Prove (or disprove) the following $$\det(qA) = q^{n} \det(A).$$ I tried disproving it with counterexamples but I could not find one.
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
2answers
55 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 ...
2
votes
2answers
40 views

closeness of a set of vectors

Is there some measure that captures the "closeness" of a set of vectors? Say I have a matrix, $$ A = \left[ \begin{matrix} 0.8 & 0.15 & 0.05 \\ 0.82 & 0.09 & 0.09 \\ 0.78 & 0.08 ...
0
votes
2answers
51 views

Determinant of complex matrix

How is the determinant of a complex matrix calculated? Is it the same algorithm as for real matrices, but the determinant itself is complex instead of real? (I was unable to find any hints with ...
0
votes
0answers
20 views

Regularity of matrix

i have the set of equations $\sum_{i=0}^n (\gamma_i i^r - \beta_i r i^{r-1}) = 0$ for $r=1,..,2n$ with given real coefficients $\gamma_i, \beta_i$ and $\gamma_n = 1$ and have to prove that the ...
1
vote
2answers
75 views

Show that a matrix has positive determinant

Let $A$ be an $n\times n$ matrix, where $a_{ii}>0$ and $a_{ij}\le 0$ for $1\le i\ne j\le n$ and also $\sum_{i = 1}^n a_{ij}>0$, show that $\det(A)>0$. I try to use the fact that ...
3
votes
2answers
81 views

Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
0
votes
0answers
34 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
2
votes
2answers
57 views

Determinant of an ill conditioned matrix

I have the following ill conditioned matrix. I want to find its determinant. How is it possible to calculate it without much error \begin{equation} \left[\begin{array}{cccccc} ...
0
votes
2answers
41 views

Is this value correct or should it be simplified?

Given that $a\neq p$, $b\neq q$, $c\neq r$, and $\begin{vmatrix} p & b & c \\ a & q & c \\ a & b & r \end{vmatrix} =0$ Then find the value of ...
0
votes
1answer
20 views

the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
2
votes
3answers
50 views

Proving linear independence of matrices

Let $A = \textrm{diag}(a_{1},a_{2},a_{3})$ where $a_{1},a_{2},a_{3}$ are distinct. I am trying to show that every diagonal $3\times3$ matrix cane be made up of linear combinations of $I$, $A$ and ...
0
votes
1answer
23 views

Does cofactor expansion generalize to complex matrices?

When finding the determinant of some $n * n$ matrix $A$ when $$\forall i,j\in\mathbb{N} ,i\leq n\land j\leq n\implies A_{ij} \in \mathbb{C}$$ Can cofactor expansion be used under the normal definition ...
0
votes
2answers
28 views

Show determinant equals 0

Ok, i've been working on the following problem and this is what I've gotten: Let $F$ be a field, let $n$ be a positive integer, and let $A,B \in M{nxn} (F)$ be matrices satisfying $B\ne 0$ and ...
0
votes
0answers
29 views

Computing characteristic polynomial of tridiagonal block matrices

I want to compute the characteristic polynomial of symmetric matrices of the form \begin{bmatrix} A & U & & & 0\\ U & B & V & &\\ & V& C& W &\\ ...
0
votes
3answers
87 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
15
votes
9answers
894 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
21
votes
2answers
1k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
1
vote
2answers
37 views

How to prove determinant is a group homomorphism and onto?.

I posted this question I am struggling with previously but it was put on hold for lack of context, I hope this is now clearer. Consider the determinant function Det: Mn($\mathcal{F}$) $\to$ ...
2
votes
2answers
91 views

Calculate $\lvert A \rvert$ if $a_{ij}=0$ if $i=j$ and $1$ otherwise [duplicate]

Let $n$ be a positive integer and let $A=[a_{ij}] \in M_{n\times n} (R)$ be the matrix defined by $a_{ij}=0$ if $i=j$ $1$ otherwise To be honest, I've only calculated determinants of matrices ...
0
votes
2answers
52 views

How find this matrix determinant value

Find the value $$ \det\left| \begin{array}{c&c&c&c&c} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & AB^2 & AC^2 & AP^2 \\ 1 & AB^2 & 0 ...
3
votes
2answers
99 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
14
votes
4answers
496 views

Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$. My conjecture is ...
4
votes
2answers
96 views

How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$ [duplicate]

Question: let the matrix $A=(a_{ij})_{n\times n},i=1,2,\cdots,n,j=1,2,\cdots,n$, and the matrix $M=(a_{ij})_{(n-2)\times (n-2)},$ mean that $$A=\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ ...
1
vote
1answer
63 views

Compute the determinant $4\times 4$

Compute the determinant: $$ A= \begin{vmatrix} 1 & 1 & a+1 & b+1 \\ 1 & 0 & a & b \\ 2 & b & a & b \\ 2 & a & a ...
1
vote
1answer
41 views

Four coplanar points in $\mathbb{N}^3$ space

Is it possible to write out natural number coordinates of four three-dimensional points $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d} \in \mathbb{N}^3$, with the following determinant zero? ...
0
votes
1answer
13 views

Evaluating Determinants using elementary operations

I'm having problem regarding the Evaluation of a determinant. Can anyone explain me if there are any rules regarding row operations? I mean which row should I evaluate first?
0
votes
1answer
24 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
2
votes
0answers
49 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
4
votes
2answers
75 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
0
votes
1answer
17 views

Taking product of cofactor with different row

Given a matrix $A=(a_{ij})_{n\times n}$, let $C_{i,j}$ be the cofactor in position $(i,j)$. By the determinant formula, we have $$\det A=\sum_{i=1}^n a_{i,1}C_{i,1}.$$ What about if we take a ...
1
vote
1answer
19 views

Determinant of matrix of linear transformation in complex vector space

Let $V$ be finite complex vector space, $a\not= 0$ an element of $V$, and $f$ linear functional on space $V$. $A: V \to V$ has definition: $A(x)= x - f(x)*a$. Find determinant of $A$.
1
vote
2answers
77 views

$\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let A be real square matrix. If $\det (A^2 - I) < 0$, then A has eigenvalue $\lambda \in (-1,1)$. How to prove this?