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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4
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1answer
120 views

$A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?

Let $A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?
1
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2answers
56 views

Find $\det$ in terms of $k$

Consider the following matrix: \begin{bmatrix} 1 & 2 & 3 \\ 2 & k-3 & 4 \\ 3 & 4 & k-4 \\ \end{bmatrix} I have the following problems: How to find ...
3
votes
2answers
30 views

system of equations when the matrix corresponding $\det(A)=\pm1$ has integers solution

I am reading a book about continued fractions and one of the theorem's proof constructs a system of linear equations and states that the matrix corresponding with the system of equations satisfies ...
0
votes
2answers
40 views

How to compute this determinant, without the Sylvester determinant theorem, [duplicate]

The problem statement is: Show that there exists numbers $a$ and $b$ such that $$det (A + sxy^*)= a+bs$$ here $A$ is an $nxn$ matrix with real entries, and $x,y\in R^n$. I've been using brute ...
7
votes
1answer
43 views

Prove that p divides to algebraic multiplicity of the eigenvalue

I need help in the following exercise of a qualifying exam: Let $A$ be a matrix of size $m$ by $m$ over the finite field $\mathbb{F}_p$ such that $\operatorname{trace}\left(A^n\right)=0$ for all $n$. ...
5
votes
1answer
34 views

link between determinant

I'm a bit confused with this determinant. We have the determinant $$\Delta_n=\left\vert\begin{matrix} 5&3&0&\cdots&\cdots&0\\ 2&5&3&\ddots& &\vdots\\ ...
2
votes
1answer
38 views

Greatest natural number which divides the determinant of a matrix

All elements of a 100 x 100 matrix ,A are odd numbers. What is the greatest natural number that would always divide the determinant of A? I have been able to show that it is always divisible by ...
1
vote
1answer
22 views

Matrix equality and singularity

If two matrices $M$ and $N$ are related as $M^2 = N^2$ and they are not singular, can it be concluded that $M=N$? If $M^2= N^2$ and $MN =NM$ also, and it is known that $M \ne N$, can we conclude ...
-1
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2answers
40 views

How do I solve this invertible matrix problem?

Given an invertible matrix $A$ such that all the elements in $A$ and in $A^{-1}$ are integers, find $|A^4|$.
1
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2answers
33 views

How does this determinant calculation work?

Given that $a_0, a_1,...,a_{n-1} \in \mathbb{C}$ I am trying to understand how the following calculation for the determinant of the following matrix follows: $$ \text{det} \begin{bmatrix} x & 0 ...
1
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0answers
12 views

Prove that $\frac{1}{[Z^{-1}]_{kk}}=\frac{\text{det}Z} {\text{det}Z_{kk}}=\text{det}Z_{kk}^{\text{SC}}$, $Z_{kk}^{\text{SC}}$ is the Schur complement

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading ...
6
votes
2answers
50 views

Help me with the result of this determinant..

$$ D = \begin{vmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 2 & 1 & 1 & \dots & 1 & 0 \\ 3 & 1 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & ...
0
votes
2answers
54 views

Determinant of a square matrix with a particular pattern

Let $A=[a_{ij}]$ be the square matrix of order $n$ whose entries are given as follows. For $1\le i,j\le n$ we have $$a_{ij}= \begin{cases} ...
1
vote
3answers
44 views

Find the value of the following $n \times n$ determinantes

Find the value of the following $n \times n$ determinantes $$\begin{vmatrix} a_1+x & x & x & \ldots & x \\ x & a_2+x & x & \ldots & x \\ x & x & a_3+x & ...
0
votes
1answer
56 views

Determine a determinant is divisible by 23 or not

Consider the fact that $25875, 46552, 41354, 48691, 95818$ are all divisible by $23$. Use this fact to determine if \begin{vmatrix} 2 &5 &8 &7 &5 \\ 4 &6 &5 &5 &2 \\ 4 ...
2
votes
2answers
43 views

Finding the determinant of anti-diagonal matrix

How would one find the determinant of an anti-diagonal matrix ($n \times n$), without using eigenvalues and/or traces (those I haven't learned yet): My initial idea was to swap the first and n-th ...
3
votes
0answers
42 views

Is there a name for this generalization of the determinant?

In the context of averaging over network paths, I arrived at a certain generalization of the determinant for an $n\times n$ square matrix $A$, that is $$D_k(A) := \sum_{(j_1,j_2,...,j_n):\,\, ...
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0answers
14 views

Determinant of almost skew symmetric matrices

Recently I was simply playing around with matrices and I discovered this: If a matrix is skew symmetric except for its middle element then it's determinant is equal to the negative of the product of ...
1
vote
1answer
12 views

Cofactor multiplied with another row

Why is it that when I add up the product of cofactors for one row and a corresponding element of any other row , the answer is 0? For example: This seems to work for all matrices but I'm unable to ...
1
vote
0answers
25 views

Determinant of a jacobian

I have the following problem.The jacobian matrix is given in the image below.I just cannot seem to figure out how they arrived at the determinant.Can anyone show the steps or elaborate the procedure? ...
6
votes
2answers
199 views

Prove that determinant of the matrix is non-zero

Given a square matrix $A$ of order $2n$ such that $a_{ii}=0$ and $a_{ij}\in\{-1,1\},\space i\neq j$, prove that $\det(A)\neq0$.
3
votes
1answer
64 views

Deducing a derivative from its evaluation at the identity

I have shown that the Frechet derivative at $\mathbf{I}$ of the determinant map is $\text{tr}\,\mathbf{H}$. In notation: $$D \det \mathbf{A}\big|_{\mathbf{I}} (\mathbf{H})=\text{tr}\,\mathbf{H}$$ ...
0
votes
0answers
14 views

Cofactors and determinant equals zero

"If detA = 0 then at least one of the cofactors must be zero". It's said the this sentence is false. But why? Why can't a cofactor be zero? The solution also adds that all of the cofactors equal 1. ...
2
votes
1answer
49 views

Given an invertible matrix $A$ such that all elements in $A$ and in $𝐴^{−1}$ are integers, find $|𝐴^4|$

I find this question very interesting, but I am having trouble figuring out how to approach the problem. I know that the $\det(A^{-1}) = 1/\det(A)$, but I'm unsure of where to go from here. If ...
1
vote
1answer
33 views

Find the eigenvalues of an unsymmetrical matrix

Eigenvalues of $\begin{bmatrix}4 & -5 & 1 \\ 1 & 0 & -1\ \\ 0 & 1 & -1\end{bmatrix}$ I usually set $0$ equal to $\det(A- \lambda I)$ to find the eigenvalues, but the book ...
0
votes
1answer
54 views

If $\mathrm{det}(M)$ satisfies [generalization of being a unit], then $M$ satisfies [generalization of having a two-sided inverse].

(I'm interested in arbitrary commutative rings, not just integral domains or PID's.) Let $R$ denote a commutative ring and $M$ denote an $n\times n$ matrix over $R$. Suppose we're interested in ...
2
votes
1answer
28 views

Find the Values of n and k for which the determinant of the Matrix M(n,k) is Singular

I have been stuck on this problem for a couple of days, I don't want the answer, but I would appreciate some help in finding it! Thanks in advance! Consider a Symmetric Square Matrix $M(n,k)$ such ...
5
votes
0answers
55 views

A determinantal equality

Mark Kac wrote a paper about asymptotics of determinants whose main diagonal is taken from a function $f$, with $-1$ on the super and sub-diagonals. Specifically, $$ D_n = \begin{vmatrix} f(1/n) ...
1
vote
4answers
50 views

$n^{th}$ determinant

Find determinant $D_n$ of matrix $$ \begin{bmatrix} 1 & 1 & \cdots & 1 & -n \\ 1 & 1 & \cdots & -n & 1 \\ \vdots & \vdots & ...
2
votes
4answers
163 views

How to show that det(A)≤1?

Let $A = (a_{ij})_n$ where $a_{ij} \ge 0$ for $i,j=1,2,\ldots,n$ and $\sum_{j=1}^n a_{ij} \le 1$ for $i = 1,2,\ldots,n$. Show that $|\det(A)| \le 1$. Should I use the definition of matrix: ...
0
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1answer
23 views

Matrix Determinants Equivalence

The left-hand side becomes A*I_n - B*0_n,n = A, correct? How can A = det(A) just from the information given?
2
votes
1answer
26 views

Determinant of infinite matrix with non-zero elements above and below diagonal

I came across this idea when studying perturbations in Q.M. Is it possible to somehow show what the determinant would be of a matrix of this form: \begin{matrix} 0 & a & 0 & \dots ...
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votes
0answers
11 views

Constructing a Minimal DFA from L = (ab + b)* ba using Brzozowski's derivatives method

How would I use Brzozowski's derivatives method to construct a minimal DFA recognizing the language defined by the rational expression: L = (ab + b)* ba
2
votes
3answers
40 views

If J is a 101×101 matrix with all entries equal to 1 and let I denote the identity matrix of order 101. Then what is the determinant of J-I?

If $J$ is a $101\times 101$ matrix with all entries equal to $1$ and let $I$ denote the identity matrix of order $101$. Then what is the determinant of $J-I$ ?
0
votes
0answers
26 views

Using the Jacobian matrix to find surface area without a change of basis.

http://mathinsight.org/parametrized_surface_area_examples In reading through the example in the above link, it's straightforward to find the surface area for a cone as follows. Find the surface ...
0
votes
1answer
16 views

Understanding Jurgen Neukirch proof that $A[b_1, \dots, b_n]$ finitely generated $\implies$ it's integral over $A$.

... Conversely, assume that the $A$-module $A[b_1, \dots, b_n]$ is finitely generated and that $w_1, \dots, w_r$ is a system of generators. THen for any element $b \in A[b_1, \dots, b_n]$, one ...
0
votes
1answer
32 views

Is there a general rule of thumb for calculating determinants?

For example if we're told matrices A and B are both 3x3 matrices with det(A)=2 and det(B)=4. I know that det(AB) = 8 Also det(3$A^2$)=108, but I don't understand how or why that is. But for ...
0
votes
1answer
25 views

Manipulation of skew-symmetric linear map

Let $\Delta$ be a skew-symmetric $n$-linear map. I have the following in my notes and I am having trouble seeing how it follows: $$ \Delta\left(\sum_{i=1}^n{e_i}, \sum_{i=1}^n{(e_i)} -e_2, ...
0
votes
3answers
38 views

How to prove that $\det(Z_{n}) = \det(Z_{n-1}) - \det(Z_{n-2})$?

I'm given an $n \times n$ matrix $Z_{n}$ over $\mathbb{N}$ of which the entry in the $x$-th row and the $y$-th column equals 1 if $|x-y| < 1 $ or $ |x-y| = 1$ and zero otherwise. I'm trying to ...
2
votes
4answers
73 views

Matrix's determinant

I've got to calculate determinant for such matrix: $$ \left[ \begin{array}{cccc} a_1+b & a_2 & \cdots & a_n\\ a_1 & a_2+b & \cdots & a_n\\ \vdots & \vdots & \ddots ...
-2
votes
1answer
28 views

compute determinant and is A invertible?

Compute the determinant of A = \begin{bmatrix} 0 & 1+i & 2 \\ -2i & 0 & 1-i \\ 3 & 4i & 0 \end{bmatrix} along the third row. Is A invertible?
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1answer
44 views

Proving the Formula for the Determinant of the Adjacency Matrix of a Complete Graph

A complete graph of $n$ nodes has an $n$x$n$ adjacency matrix $A_{ij}$ such that $$ a_{ij} = 0 \text{, if } i = j \\ a_{ij} = 1 \text{, if } i \ne j $$ i.e. there are 0s down the diagonal and 1s ...
2
votes
2answers
37 views

Find all the solutions for the equation det(AB) = 0

So i'm trying to solve this problem from Jin Ho Kwak's book on linear algebra: Find all the solutions for the equation $\det (AB)=0$ where \begin{align}A&=\begin{pmatrix}x+2 &3x\\3 ...
2
votes
1answer
325 views

How do I prove that an anti-symmetric matrix $A$ is not invertible?

$A$ is a square anti symmetric matrix with dimension $n\times n$. It is known that $n$ is an odd number. Prove that $A$ is not invertible. How do I prove this? any hints please?
0
votes
2answers
68 views

is determinant of A times A transposed bigger than or equal to zero?

We have an m by n matrix A of real numbers where n is bigger than m. Prove that determinant of A times A transposed is bigger than or equal 0.
2
votes
2answers
33 views

Calculate determinant with induction

I need to prove the following, with induction to every $1 \leq n$: $$D(a_1,...,a_n) = \left| \begin{array}{ccc} a_1+x& a_2 & a_3 & \cdots & a_n \\ a_1& a_2+x & a_3 & ...
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2answers
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Prove that $\alpha=\beta=\gamma$

Q. Let $x,y,z \in \Bbb R-\{0\}$ and $\alpha,\beta,\gamma \in \Bbb C$ such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z$, then prove that $\alpha=\beta=\gamma$. My ...
2
votes
1answer
54 views

How to calculate a determinant of a 2x2 symmetry block matrix?

I'd like to calculate the determinant of the matrix: $$ \begin{pmatrix} -A & B^\star \\ -B & A^\star \\ \end{pmatrix} $$ $A$, $B$ are $L\times L$ complex ...
0
votes
1answer
22 views

Linear independence, if $ |a_{kk}| > \sum_{i=1, i \neq k}^{s} |a_{ik}| $

I found task written below, but I cannot prove it. Given is system of $s$ vectors ($a_i = (a_{i1}, a_{i2}, \dots a_{in})$ for $i = 1, \dots, s$), where $s \leq n$. Prove that, if for all $1 \leq ...
-1
votes
3answers
61 views

Bounds on determinant [closed]

I was just curious about the bounds on the determinant of a 3x3 matrix whose elements take values between 0 and 5. I believe this bound is around +-1040