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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1answer
39 views

Does there exist a simple solution to the following eigenvalue problem

I am looking for the values of $Z$ for which the determinant of the following $N$-dimensional matrix vanishes: \begin{equation} \begin{bmatrix} N(1-Z) & N-1 & N-2 & \cdots & \cdots ...
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0answers
25 views

What is the fastest method for finding the determinant of any square matrix?

There are several methods to find the determinant of a matrix. What is the fastest method to fastest for finding the determinant of any square matrix. Any square matrix being a matrix that is ...
1
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0answers
39 views

Hypermatrices, hyperdeterminants and Grassmannians.

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ ...
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4answers
46 views

Find matrix from Eigenvectors and Eigenvalues

A matrix $A$ has eigenvectors $v_1 = \left( \begin{array}{c} 2 \\ 1 \\ \end{array} \right)$ $v_2 = \left( \begin{array}{c} 1 \\ -1 \\ \end{array} \right)$ with corresponding ...
1
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0answers
23 views

Powers of coefficients divide the resultant

Let $f(x)=a_0x^n+a_1x^{n-1}+\dots+a_n$, $g(x)=b_0x^m+b_1x^{m-1}+\dots+b_m$, with coefficients in a field. Prove that $a_0^mb_m^n$ divides the resultant of $f(x)$ and $g(x)$. I have written the ...
3
votes
4answers
382 views

When a determinant is zero

Is it true that if $C$ is a square matrix of size $n$ and $\det(C) = 0,$ then $C^n = O_n$ or the $0$ matrix? If yes, then why is that? I know that the reverse is obviously true, so I wondered if ...
0
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1answer
50 views

Calculating the Jacobian of inverse functions

The task is this: given the following pair of functions: \begin{cases} u = e^x cos(y) \\ v = e^x sin(y) \end{cases} Determine the inverse functions, and compute the Jacobian of the inverse functions ...
2
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0answers
38 views

Is the Cone over Grassmannian manifold a determinantal variety?

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. ...
1
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1answer
55 views

Determinant of the symmetric part of a matrix.

Define the symmetric part of a matrix $A$ as: $$ A^+ := \frac{A+A^t}{2}. $$ Is there a formula relating the determinants of $A$ and $A^+$? Thanks!
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0answers
65 views

Invertability of a matrix

$\newcommand{\AA}{\mathbf{A}} \newcommand{\Tr}[1]{\operatorname{Tr}\left[#1\right]}$ I have a problem that I suspect there is a “relatively” simple answer to but it is currently eluding me. I am ...
6
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2answers
371 views

How can I quickly find the determinant of this matrix

$$ \begin{vmatrix} 14 & 2 & 1 & 3\\ 31 & 4 & 5 & 6\\ 26 & 3 & 7 & 4\\ 10 & 1 & 3 & 2\\ \end{vmatrix} ...
0
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1answer
20 views

Characteristic Polymonmial 4x4 Matrix

I have to find the characteristic polynomial to find Jordan normal form. I chose to solve this via column expansion on the first determinant, and then row expansion in the inner determinant. But ...
0
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1answer
39 views

LU Decomposition

I'm having trouble understanding which answer is correct. I'm currently reading a paper: lecture 12 - They give the following example: Let: $$ A = \begin{bmatrix} 1&2&3 \\ 2&5&12 ...
5
votes
4answers
319 views

Decompose this matrix as a sum of unit and nilpotent matrix.

Show that the matrix $A=\begin{bmatrix} 1 & 0 \\ 2 & 1 \\ \end{bmatrix}$ can be decomposed as a sum of a unit and nilpotent matrix. Hence evaluate the matrix ...
0
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1answer
35 views

is the Jacobian Determinant continuous

Is the Determinant of the Jacobian a continuous function? i.e. $$f:\mathbb{R}^n \rightarrow \mathbb{R}^n $$ $$ \forall \varepsilon >0 \quad \exists \delta >0 : |x-x_0 |<\delta ...
0
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0answers
20 views

Question on a proof concerning a Sylvester matrix and the roots of polynomials

For clarity here, $R(f,g)$ is the determinant of a Sylvester matrix. Also, the author writes a polynomial as $f(x) = a_{0}x^n + ... + a_{n}$. Whats the reason as to why $\lambda = 1$? I don't ...
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2answers
35 views

Inequality involving Hadamard's inequality

Let $A$ be matrix in $\mathbb{R^{m \times n}}$. Let $A$ and $B$ be quadratic submatrices of $M$ such that $\det(A)< \det(B)$. Does this imply $\prod_{i=1}^n \|A^i\| < \prod_{i=1}^n \|B^i\|$ ...
6
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3answers
52 views

Calculation of determinant

Is there any easier way to make sure the determinant of the following matrix is n (the dimension of square matrix)? $ \begin{vmatrix} 1 & -1 & -1 & -1 & \cdots & -1 \\ 1 ...
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0answers
36 views

Characteristic Polynomial Calculation

I have a problem in my homework in which I have to find the characteristic polynomial of the following matrix: I know the final solution is: However, my answer keeps getting wrong whenever I ...
0
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2answers
28 views

Inverse of matrix sum

I found on the Wikipedia page "Determinant" the following property: For any invertible $m \times m$ matrix $X$, $\det(X + AB) = \det(X) \det(I_m + BX^{-1}A)$. Is this true? If so, how is this ...
4
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1answer
81 views

Uniqueness of determinant

In Artin Algebra 2nd edition page 22, the author proved the uniqueness of determinant by saying that any matrix $A$ can be written in reduced row-echelon form $A'$: $A'=E_1\cdots E_kA$ where $E_i$ are ...
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0answers
31 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
0
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1answer
27 views

Finding the real irrational root of a cubic polynomial?

I just wanted to check if anyone can see a simpler way to solve this. Because I am not looking forward to using the cubic formula to solve it! $$ det(\lambda-AI) = \left| \begin{array}{ccc} \lambda + ...
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0answers
33 views

Proof for Determinants using Laplace and induction.

Matrix $A = (a_{ij}) \in M (n \times n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ I need to prove that det(A)=det(B). I thought induction might be one solution, but I don't know how to apply the ...
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0answers
25 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
0
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1answer
34 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
5
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3answers
87 views

Determinant of the inverse matrix [duplicate]

I'm seeking for a proof of the following: Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals: $$\left|A^{-1}\right|=|A|^{-1} $$ I don't know where to begin the proof. Any ...
1
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1answer
50 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
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0answers
44 views

Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
1
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2answers
36 views

Determinant by nullifying

I am supposed to calculate the value for the determinant of this matrix. I didn't know what to do, so I looked up for the sample solution, which I don't understand. $$\left|\begin{array}{ccc} 18 ...
2
votes
1answer
55 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
1
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2answers
60 views

How many solutions exist for a matrix equation $A^2=I$?

Let $A$ be a square matrix of order three or two, and $I$ be a unit matrix. How many solutions are possible for the equation $$A^2=I$$? In case the solutions are infinite, or very large, how do I ...
2
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0answers
30 views

Log concavity/convexity of a determinant

I was wondering if anyone would be able to help me determine whether the following quantity is log concave or not with respect to $\alpha$? $$\left[\det(\textbf Y^\top \textbf P \textbf G \textbf ...
1
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1answer
43 views

Determinant and matrix power

I was wondering if there is a relation between the determinant of a matrix and the determinant of its powers. I mean I am looking for something like $$ \det (A^k) = f(\det(A), k). $$ A few check I ...
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0answers
21 views

Transformation into a field with the result being a multiple of the determinant

Let $K$ be a field, $n \in N$ and d: $M_{n,n}(K) \to K $ an homogeneous and skew invariant transformation where $M_{n,n}(K)$ are the matrices over the field. Show that there's a $d$ with $d = c * ...
0
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1answer
27 views

Given $\det(A)$ and $\det(B)$, is my calculation of $\det(-2B^T B A)$ correct?

Suppose $A$ and $B$ are $3 \times 3$ matrices with $\det(A) = -2$ and $\det(B) = -1$. What is the determinant of $C = -2 B^T B A$? I know that $$\det(A^T) = \det(A) \qquad \det(AB) = \det(A) ...
1
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1answer
32 views

Determinant of 3 points.

I have $P=(p_1,p_2)$ and $Q=(q_1,q_2$) two points in $\mathbb R^2$, $P\ne Q$, and $R=(r_1,r_2)$ another point. What means the following determinant? $$\Delta (P, Q, R)= \begin{vmatrix} ...
1
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1answer
48 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
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0answers
29 views

Determinant of matrix n x n [duplicate]

How to calculate $det\begin{bmatrix}1 & x_1 & x_1^2 \dots x_1^{n-1} \\ 1 & x_2 & x_2^2 \dots x_2^{n-1} \\ \\ 1 & x_n & x_n^2 \dots x_n^{n-1}\end{bmatrix}$?
1
vote
1answer
35 views

Determinant of 2 transpose matrix A and B.

Can you show me why $\det(A^T B^T) = \det(A)\det(B^T) = \det(A^T)\det(B)$ ? im really having a hard time finding its properties. i dont know what to search. please help.
2
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2answers
31 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
0
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1answer
49 views

Correct proof of $\det(X+iY) \det(X-iY)>0$?

Can someone please look over my proof below as to why $\det(X+iY) \det(X-iY)>0$ for real matrices $X,Y$, such that $\det(X+iY)$, $ \det(X-iY)$ not both the zero, and tell me if it's correct ? My ...
4
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1answer
31 views

linearly independent and determinant

This question says a matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ where $a_{ij}$ are real numbers. I need to prove that $\det|A|=ad-bc\neq0 \iff $the columns are linearly independent. ...
2
votes
2answers
89 views

Block Matrix Determinant Proof

I am trying to solve the determinant of a Block matrix $$\begin{bmatrix}A-Ia&B\\B &A-Ib \end{bmatrix}$$ where a and b are integers and I is an identity matrix, A and B are square. ...
0
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1answer
15 views

Express the vector $b=2i-3j+5j$ in terms of these set of three vectors

The three vectors are: $$a_1=i+j+k$$ $$a_2=i-j$$ $$a_3=i+j-2k$$ I have been asked to express the vector $b=2i-3j+5j$ in terms of the three vectors above like: $b= \alpha a_1+\beta a_2+\gamma a_3$. ...
0
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1answer
32 views

What method is used to find the determinant of this $4 \times 4$ matrix?

This is a pre-solved example in my book, I don't understand how they solved it. What method is used? Find the determinant of $A = \begin{bmatrix} 0 & 1 & 0 & 2\\[0.3em] -1 ...
2
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0answers
52 views

Determinant of a sum

We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is ...
3
votes
1answer
32 views

Find the value of the expression-

Consider a matrix $A=\begin{bmatrix}3 & 1\\-6 & -2\end{bmatrix}$, then $(I+A)^{99}$ equals ? So how can I expand this ? The solution paper gives the answer as $I+(2^{99}-1)A$
2
votes
1answer
47 views

How to prove these two statements?

Let A,B,C,D be real matrices (not necessarily square) such that $$A^T=BCD$$$$B^T=CDA$$$$C^T=DAB$$$$D^T=ABC$$ For the matrix S=ABCD, prove that $$S^3=S$$ and $$S^2=S^4$$ My little brother got this in ...
2
votes
1answer
40 views

Maximum value of $f(x) = \log_{(\tan x + \cot x)}(\det A)$ for a diagonal matrix $A$

If $$A =\begin{pmatrix} d_1 & 0 & 0 & 0 \\ 0 & d_2 & 0 & 0\\ 0 & 0 & d_3 & 0\\ 0 & 0 & 0 & d_4\\ \end{pmatrix}$$ ...