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.
0
votes
4answers
120 views
Evaluation of a specific determinant.
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.
$$A_2
\begin{pmatrix} 1& 1\\
1& 2
\end{pmatrix};
A_3 = ...
1
vote
1answer
45 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
2
votes
3answers
121 views
On integral of a function over a simplex
Help w/the following general calculation and references would be appreciated.
Let $ABC$ be a triangle in the plane.
Then for any linear function of two variables $u$.
$$
\int_{\triangle}|\nabla ...
0
votes
1answer
50 views
Calculate the determinant of a multilinear operator
How to calculate the determinant of a multilinear operator? Is it something different from the determinant of the linear operator? Thanks.
1
vote
1answer
59 views
Elementary Linear Algebra
What does it mean when someone says "find a fundamental set of solutions for the system y' $=A$ y"?
That is, the system
$$ {\bf{y'}} =A {\bf{y}}. $$
11
votes
5answers
261 views
Show determinant of matrix is non-zero
I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help ...
1
vote
2answers
31 views
determinants of 2 matrices with given property
I have two $3\times3$ integer matrices $A$ and $B$ such that $AB=A+B$. I need to find all possibe values of $\det(A-E)$, where $E$ denotes the identity matrix. Any help is appreciated.
3
votes
1answer
36 views
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it.
Thanks for ...
1
vote
1answer
34 views
Computing Resultant
The resultant of two polynomials is defined as the determinant of the Sylvester matrix. If the polynomials are of degree $n$ and $m$, than the Sylvester matrix will be of dimension
$(m+n)\times ...
3
votes
2answers
29 views
Linear Algebra determinant and rank relation
True or False?
If the determinant of a $4 \times 4$ matrix $A$ is $4$
then its rank must be $4$.
Is it false or true?
My guess is true, because the matrix $A$ is invertible.
But there is ...
1
vote
1answer
40 views
A linear algebra problem - matrix equation
Let $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n$ column vectors, each with the same $n$ components. So:
\begin{equation}
\mathbf{v}_i = \left[\begin{array}{c}v_i\\ v_i \\\vdots \\ ...
0
votes
3answers
29 views
$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form
Prove that every $n$-linear alternating form on a vector space of dimension
less than $n$ is the zero form.
4
votes
1answer
71 views
When does a matrix $A$ with ones on and above the diagonal have $\det(A)=1$?
What conditions, if they're even necessary, must be placed on $\star$ so that the matrix
$$ \begin{pmatrix} 1 & & \huge{1} \\ & \ddots & \\ \huge{\star} & & 1 \end{pmatrix}, ...
4
votes
2answers
84 views
Determinants: A Special Condition
Under what conditions is
$$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$
just curious.
0
votes
1answer
152 views
Inequalities involving determinants and minors of positive definite matrices
Lately I've been dealing with positive definite matrices in my research (in the context of them being covariance matrices), and, am wondering if anyone knows of a comprehensive list of inequalities ...
0
votes
2answers
35 views
Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices
Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
5
votes
3answers
70 views
How to show that $\det(A+I)\ne 0$
How to show that for any skew symmetric real matrix $A$, we have $\det(A+I)\ne 0?$
Where to begin? I'm looking for some clue only.
4
votes
2answers
116 views
Multiplication of determinants
Show that for any vectors $\bf{a}$,$\bf{b}$,$\bf{c}$,$\bf{u}$,$\bf{v}$,$\bf{w}$ in $\mathbb{R}^3$,
...
7
votes
3answers
106 views
Matrix Determinant Identity
I have come across an observation about the determinant of a matrix, but I don't know how to prove it in general. Let me demonstrate it through an example.
$$
\begin{align}
\left|
\begin{matrix}
1 ...
1
vote
0answers
36 views
Determinant of a matrix with variables in it
Assuming that $z \neq 0$, compute the determinant $d_n(z) = \det D_n \left(1, z, 1 - \frac{1}{z^2} \right)$, where $z$ is a complex variable. In particular, compute the value $d_n(\sqrt{2})$.
...
131
votes
9answers
8k views
What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
1
vote
1answer
69 views
How to calculate the determinant of a matrix with …
How to calculate the determinant using Laplace?
$$
\det \begin{bmatrix}
-t & 0 & 0 & \dots & 0 & a_1 \\[0.3em]
a_2 & -t & 0 & \dots & 0 ...
1
vote
2answers
47 views
How to calculate the determinant of a matrix using Laplace?
How to calculate the determinant using Laplace?
$$
\det \begin{bmatrix}
0 & \dots & 0 & 0 & a_{1n} \\[0.3em]
0 & \dots & 0 & a_{2,n-1} & ...
12
votes
3answers
197 views
Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$
I have to prove/disprove this:
If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$
I believe it is true but I can not think of a direct way to prove it.
Any ...
1
vote
2answers
85 views
Simplest way to calculate a determinant [duplicate]
The big $1$'s here just mean that the lower and upper triangular entries are all $1$'s. The trace entries are all zero. The matrix is for a general $n\times n$ matrix of this form. I'm trying to ...
3
votes
0answers
32 views
What is the limit $\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le m/n\le r/s}x^my^n$?
Let $S=[0,1)^2$ and $m,n$ are positive integers and $p/q,r/s$ are positive rationals with $p/q<r/s$. What is the limit
$$\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le ...
0
votes
3answers
58 views
Characteristic value or eigenvalues and determinant
I am having semester in linear algebra. And have recently got acquainted to eigenvalues.
What is the relation between eigenvalues and determinant? Going through answers of some questions I found ...
0
votes
1answer
50 views
Linear algebra determinants
I have tried to solve this problem but I don't have an idea how to begin, any hints?
For any vector $x$ in $\mathbb{R}^n$ let $(x,x) =\sum\limits_{i=1}^n x_i^2 $ . Let $A$ be a matrix of size $n ...
3
votes
1answer
80 views
Skew symmetric matrix decomposes
I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $ R $ such that $ R^T A R = M$, ...
1
vote
1answer
65 views
prove that determinant is a quadratic form
let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$.
prove that $q(v)=\det(v)$ is a quadratic form.
I tried to prove that ...
2
votes
1answer
72 views
Determinant problem
I'm stuck in this question:
How calculate this determinant ?
$$\Delta=\left|\begin{array}{cccccc}
1&2&3&\cdots&\cdots&n\\
n&1&2&\cdots&\cdots& n-1\\
...
1
vote
2answers
51 views
Product of two matrices equals zero
If the product of two $n \times n$ matrices $A$ and $B$ is zero ie: $AB = 0$
Then either $\det(A)$ or $\det(B)$ must be zero.
What additional conditions on $A$ and $B$ would be sufficient ? Clearly ...
2
votes
2answers
43 views
Characteristic and minimal polynomial of a special matrix
$H = \begin{bmatrix}
1 & w^{-1} & w^{-2} & ... & w^{1-n}\\
w & 1 & w^{-1} & ... & w^{2-n} \\
w^{2} & w^1 & 1 & ... & w^{3-n} \\
... & ... & ...
4
votes
2answers
45 views
Calculate the determinant when the sum of odd rows $=$ the sum of even rows
I have came across this interesting question in linear algebra and I couldn't know for sure the answer.
Given a matrix $A \in M_{n \times k} (\mathbb F)$, The sum of odd rows of $A$ $=$ the sum of ...
-1
votes
3answers
41 views
Questions about matrices and determinants - constant variable multiplication
Is this matrix
$$
M = \begin{bmatrix}
a & -a & a \\[0.3em]
-a & -a & -a \\[0.3em]
a & a & a
\end{bmatrix}
$$
the same as:
$$
M = ...
0
votes
0answers
47 views
Generalizing formula for calculating determinant of specific matrix
There is a similar question like this. And this is extension of this question
How can we calculate the determinant of this $\,pn-1\times pn-1\,$ matrix. I have tried at my best level, and still am ...
0
votes
2answers
108 views
Computing determinant of this matrix
I have a very specific kind of matrix and I have to find the formula to find the determinant of these matrix.
a(i,j)=a if(i==j) and a(i,j)=0 if(floor(i/2)=floor(j/2) and i!=j) and n is odd
$$ ...
2
votes
1answer
45 views
Solving linear equations with Vandermonde
Given this:
$$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
2
votes
2answers
55 views
Calculating the determinant of this matrix
Given this (very) tricky determinant, how can we calculate it easily?
$$\begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta & \alpha ...
10
votes
7answers
248 views
Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?
I saw the following statement in my homework and we are asked to make use of the statement:
If $S$ is a symmetric matrix then
$$\det(I + S ) = 1 + \operatorname{trace}(S).$$
However, I am not ...
2
votes
1answer
106 views
Determinant of matrix?
How can we calculate the determinant of this $\,pn\times pn\,$ matrix. I have tried at my best level, and still am not able to come up with a solution. The matrix $a_{ij}$ entry is defined as
$$
...
0
votes
4answers
41 views
Divide and Conquer matrices to calculate determinant.
Do the determinant of a matrix equal to the determinant of submatrices?
$$
det\begin{pmatrix}
a_{11} & a_{12} & a_{13} & \dots & a_{1k} \\
a_{21} & a_{22} & a_{23} & ...
1
vote
1answer
46 views
a problem on solving a determinant equation [duplicate]
Let $a$ be a real number. What is the number of distinct real roots of the following
$$\left| \begin{array}{ccc}
x & a & a & a \\
a & x & a & a \\
a & a & x & a \\
...
3
votes
1answer
90 views
Different form of determinant, does it make mine wrong?
Calculate the determinant of the following $(n+1) \times (n+1)$ matrix:
$$A = \pmatrix{1 & 1 & 1 & 1 &\cdots & 1 \\ 1 & a_1 & 0 & 0 &\cdots & 0 \\ 1 ...
2
votes
1answer
52 views
Simple/Concise proof of Muir's Identity
I am not a Math student and I am having trouble finding some small proof for the Muir's identity.
Even a slightly lengthy but easy to understand proof would be helpful.
Muir's Identity
$$\det(A)= ...
1
vote
1answer
42 views
Determinant is correct but wrong when I try and check it
I have to work out the determinant of the $(n \times n)$ matrix
$$A = \pmatrix{x & y & 0 & 0 &\cdots & 0 \\ 0 & x & y & 0 &\cdots & 0 \\ 0 & 0 & x ...
1
vote
1answer
26 views
Problem related to a complex matrix
I am stuck on the following problem:
Let $P$ be a $2 \times 2$ complex matrix such that trace $P=1$ and $\det P=-6.$ Then
trace $(P^4-P^3)=?$
Can someone point me in the right direction? ...
1
vote
2answers
68 views
Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$
Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$
Ran into trouble with a proof for linear algebra. $C$ is the cofactor matrix of $A \in \mathbb{R}^{n\times n}$, and I'm not sure how to ...
19
votes
4answers
358 views
Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected
Math people:
In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
3
votes
1answer
50 views
Find the smallest square matrix in which some objects fit following some rules
I have to put some objects in a matrix. The data of these objects is given in another matrix in which each line contains an object, and the first column represents its width, and the second its ...
