0
votes
1answer
46 views
0
votes
1answer
61 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 ...
4
votes
2answers
53 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 ...
2
votes
3answers
45 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
159 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 ...
12
votes
0answers
109 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
153 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
376 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
25 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
22 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
225 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 ...
5
votes
1answer
83 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
37 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
54 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}= ...
2
votes
2answers
68 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
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 ...
0
votes
1answer
66 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 & ...
2
votes
0answers
52 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
38 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
40 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
68 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 \\ ...
3
votes
2answers
75 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
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
67 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
37 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
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
123 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: ...
1
vote
1answer
39 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 ...
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 ...
2
votes
2answers
45 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
55 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
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
3answers
51 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
33 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 ...
15
votes
9answers
914 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 ...
2
votes
2answers
93 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 ...
3
votes
2answers
104 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 ...
13
votes
4answers
517 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
109 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}\\ ...
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 ...