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Questions tagged [determinant]

Questions about determinants: their computation or their 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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30 views

find the bmatrix $\det(x,1,0,0,\cdots)$

This problem comes from an advanced algebra book. He is not allowed to use the Jordan type of processing. He can only use elementary transformation knowledge to solve it. I haven't solved it for a ...
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1answer
30 views

Inequality w.r.t determinant of a non-negative definite matrix.

I am reading a paper where the author mentioned the following property without proof. Neither can I prove it nor can I find the proof in various textbooks. For any non-negative definite (i.e. ...
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0answers
14 views

prove a bilinear form is nondegenerate

In an exercice, I have the following: $B(M,N)=\frac{1}{2}Tr(M\tilde{N})$ where $B:M_2(\Bbb C)^2 \to \Bbb C$ and $\tilde{N}=^t(com N)$ Prove B is nondegenerate. I tried to prove that $M\in M_2(\Bbb ...
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1answer
36 views

How to evaluate this three diagonal determinant?2019

Can someone give me a hint how to solve \begin{align*} |A|=\begin{vmatrix} x & 1 & 0 & 0 & \cdots & 0 & 0 \\ n-1 & x & 2 & 0 & \cdots & 0 & 0 \\ 0 & ...
3
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1answer
54 views

Let $A\in M_{n \times k}(\mathbb{R})$. Show that $\det(A^TA)=\sum \det(B^2)$ where the sum runs through all $k \times k$ submatrices $B$ of $A$.

Let $A\in M_{n \times k}(\mathbb{R})$. Show that \begin{equation} \det(A^TA)=\sum \det(B^2) \;\;\;\;\;\;\;(*) \end{equation} where the summation runs through all $k \times k$ submatrices $B$ of $A$ ...
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2answers
18 views

On the signature of a quadratic form

Prove that the determinant in $M_2(\Bbb R)$ is a quadratic form of signature $(2,2)$. I found the first part: the symmetric bilinear form $$B(M,N)=\frac{1}{2} ( \det(M+N) - \det (M) -\det (N) )$$ ...
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1answer
49 views

Rewrite $(\det A)^{1/n}=\min\left\{\frac{\mathrm{tr}(AC)}{n}:C \in \Bbb{C}^{n×n},C>0,\det C=1\right\}$ in terms of $\frac{\rm{tr}(CAC)}{n}$

Given $$ (\det A)^{1/n} = \min \left\{\frac{\operatorname{tr}(AC)}{n} : C \in {\Bbb C}^{n \times n}, C > 0, \det C = 1\right\}. \label1\tag1 $$ Question Show that the formula can be rewritten as $...
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28 views

Given $h$ and bases $\mathcal A, \mathcal B$ , find the representation matrices $A$,$B$, and the matrix $P$ such that $B=PAP^{-1}$.

For the homomorphism $h: \mathcal P_2 \rightarrow \mathcal P_2$ given by $1 \mapsto 3$, $x \mapsto 2x-1$, $x^2 \mapsto x^2-x-1$ (i) Find the matrix $A=Rep_{\mathcal{A,A}}(h)$ for basis $\mathcal ...
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37 views

Problem in showing that this determinant is greater than zero.

In one class the teacher showed us one problem about determinants that himself could not solve, and I tried solve the problem too, but I didn't had success even in the $3$ by $3$ case, I'm very ...
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1answer
13 views

Singular points of a matrix when the entries are restriced to a Lie Group

Let $\mathsf{SO}(3)$ be the set of $3 \times 3$ rotation matrices. Let $R\in\mathsf{SO}(3)$ and $r_{ij}$ represent the entry of $R$ sitting at the $i^{th}$ row and $j^{th}$ column, i.e., $$ R \in\ \...
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2answers
135 views

Determinant is linear as a function of each of the rows of the matrix.

Today I heard in a lecture (some video on YouTube) that the determinant is linear as a function of each of the rows of the matrix. I am not able to understand the above statement. I know that ...
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2answers
32 views

the value of a determinant

Let be a polynomial function $P\in \mathbb{R}[X]$.If I divide $P$ by $(x-1)(x-2)(x-3)(x-4)$ I get a remainder without "free term" ( like $ax^{3}+bx^{2}+cx$ ) I have to calculate the determinant: $$\...
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10 views

The kronig penney model determinant expansion [on hold]

How to find the expansion of this determinant 4*4 determinant
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2answers
41 views

Given that $A+B$ is invertible, prove or disprove $A(A+B)B = B(A+B)A$ as well as show that $A(A + B)^{-1}B = B(A + B)^{-1}A$

I have tried coming up with a counterexample for the first one, but it has worked each time so my intuition is that the first statement is true. I've tried using the fact that $\det(A+B)$ does not ...
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1answer
42 views

If $A\in\Bbb R^{3\times3}$ is an invertible matrix such that $A^2=A$, then $\det(2A)=8$

True or false? "If $\pmb{A\in\Bbb R^{3\times3}}$ is an invertible matrix such that $\pmb{A^2=A}$, then $\pmb{\det(2A)=8}$". True. Proof. We start from $\det(2A)$. Then, $2^3\det(A)$. But we know ...
3
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1answer
106 views

Determinant of a matrix with positive diagonal entries is greater than 1

Let $A$ be a $n\times n$ matrix with entries on its diagonal are positive and other entries are negative with sum of entries in every column is 1. Prove that $\det(A) > 1.$ I got no idea to begin ...
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1answer
12 views

Lower Dimensional Volume under Transformation

it is a well known fact that if $K$ is a measurable set in $R^n$ (we can restrict to convex bodies if you like), and $T$ a linear transformation then $$|TK|=|detT||K|.$$ If $K$ is not $n-$dimensional ...
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1answer
39 views

Problem with determinants and polynomials $\left|\begin{smallmatrix}P(a)&&Q(a)&&R(a)\\P(b)&&Q(b)&&R(b)\\P(c)&&Q(c)&&R(c)\end{smallmatrix}\right|=1$

Let $P,Q,R$ be polynomials of degree less than 2 and $a,b,c \in \mathbb{C}$ such that $D(a,b,c)=\begin{vmatrix}P(a)&&Q(a)&&R(a)\\P(b)&&Q(b)&&R(b)\\P(c)&&Q(c)&...
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1answer
23 views

Trace and Det of Laplacian on the rectangle

I consider the eigenvalue problem $\Delta \varphi = \lambda \varphi$ with the Dirichlet boundary condition $\varphi|_{ \partial \Omega}=0$ on the rectangle $\Omega= [0,l] \times [0,m]$. By using ...
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31 views

What does it mean when $\mathrm{det}(I-Q^2)=0$ where $Q$ is Toeplitz?

Assume $Q$ is a general Toeplitz matrix. Under what conditions can we make sure $$\mathrm{det}(I-Q^2)\neq 0?$$ Let's denote the determinant by $|\cdot|$. We can show that $$|I-Q^2| = |I-Q||I+Q|\...
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Matrix determinant simplification

Find the determinant of $$A=\begin{bmatrix} I_{N \times N} & Q_{N \times N} \\ Q_{N \times N} & I_{N \times N} \end{bmatrix},$$ where $$Q=\begin{bmatrix} q_1 & q_2 & q_3 &...
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2answers
29 views

The numerical value of this determinant is?

For positive numbers $x$, $y$ and $z$, the numerical value of the determinant \begin{vmatrix} 1 & \log_xy & \log_xz \\ \log_yx & 1 & \log_yz \\ \log_zx & \log_zy & 1 \\ \...
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1answer
39 views

Expressing Determinant as Dot Product

Let $u,v,w \in \mathbb{R}^4$ and define the linear function by $F(x)=det[x \ u \ v \ w]$ for all $x \in \mathbb{R}^4$. Prove that there is a vector $z$ in $\mathbb{R}^4$ such that $T(x)=z \cdot x$ (...
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2answers
35 views

Number of real values of x that satisfy this determinant?

How many real values of $x$ satisfy the following determinant? $$ \begin{vmatrix} x & 3x+2 & 2x-1 \\ 2x-1 & 4x & 3x+1 \\ 7x-2 & 17x+6 & 12x-1 \\ \end{vmatrix} = 0 $$ If ...
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2answers
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What is 3D volume of three vectors in $n$ Dimensional Space

We have three linearly independent vectors $v_1=(v_{11} , ... , v_{1n})$ , $v_2=(v_{21} , ... , v_{2n})$ , $v_3=(v_{31} , ... , v_{3n})$. We want to calculate the 3D volume of the Parallelepiped made ...
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1answer
37 views

Property of matrix determinants

So, recently my tutor showed through a couple of examples that if you take $2$ matrices say $A_{\;3 \times 2}$ and $B _{\;2 \times 3}$ then we will either have $\det. (AB) =0$ or $\det . (BA)=0$ and ...
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4answers
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How to find $A^{100}$ where $A$ is a matrix with trigonometric entries? [duplicate]

How to find $A^{100}$ where $A=\left(\begin{matrix}\cos \theta & -\sin \theta\\ \sin \theta & \cos \theta\end{matrix}\right)$. ? (Given$\theta=\frac{2\pi}{7}$) I am a highschool student and ...
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1answer
28 views

Determinant with trigonometric functions [closed]

If $$ \begin{vmatrix} \sin 2x & \cos^2x & \cos 4x \\ \cos^2x & \cos2x & \sin^2x \\ \cos^4x & \sin^2x & \sin 2x \\ \end{vmatrix} = a_0 + a_1\sin x + a_2\sin^2x +\cdots+ ...
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2answers
62 views

Show that a determinant equals the product of another determinant and a polynomial function without calculating [closed]

Show without calculating the determinant, that $$ \det\left(\begin{bmatrix} a_{1}+b_{1}x & a_{1}-b_{1}x & c_{1}\\ a_{2}+b_{2}x & a_{2}-b_{2}x & c_{2}\\ a_{3}+b_{3}x &...
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1answer
86 views

Determinant of antidiagonally constructed matrix

Let $A_n$ be a matrix, odd dimension $n \times n$, constructed from the sequence of natural numbers in such a way that we begin the sequence from the upper left corner and next numbers are inserted ...
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Proving determinant

$$\det \begin{pmatrix} 0&x_1&x_2&\ldots&x_n\\ x_1&a_{11}&a_{12}&\ldots&a_{1n}\\ x_2&a_{21}&a_{22}&\ldots&a_{2n}\\ \ldots&\ldots&\ldots&\...
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2answers
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How to find the determinant of this matrix? (A spherical-Cartesian transformation Jacobian matrix)

I meet a difficult determinant question as the followings: $$ \text{Matrix A is given as:} $$ $$ A=\begin{bmatrix}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial\theta}&\frac{\partial ...
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How does the way we find the inverse of a matrix work? [closed]

The inverse of a matrix can be calculated as the transpose of the cofactor matrix divided by the determinant. How could it be shown that this calculation really works and indeed produces an inverse ...
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$B=((b_{ij}))$ where $b_{ij}=1 \, \forall i,j$ and $A=(a-b)i_n+bB$ Prove that $|A|=(a-b)^{n-1}[a+(n-1)b]$ [duplicate]

I have the following problem which I do not know how to solve. I found the statement to be a bit confusing. Here's the problem: If $B=((b_{ij})) \in M_{n\times n}$ where $b_{ij}=1 \, \forall i,j$ ...
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1answer
43 views

What is the “antonym of determinant” that does $f\left(\begin{bmatrix}a & b\\c & d\end{bmatrix}\right)=ad+cb$ called?

I'm looking for the name of the function $f\left(\begin{bmatrix}a & b\\c & d\end{bmatrix}\right)=ad+cb$ which I had encountered a long ago on the internet. It is the same as determinant, ...
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3answers
59 views

How to solve this $n \times n$ determinant [closed]

Find the determinant: $$\begin{vmatrix} 0 &1&0&0 &\cdots& 0\\ 0 &0&1&0 &\cdots& 0\\ 0 &0&0&1 &\cdots& 0 \\ \vdots & \vdots &\...
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1answer
35 views

Area of Parallelogram in $\mathbb R^n$

Let $\{u,v\}\subset\mathbb R^n$ be linearly independent. Then $u$ and $v$ induce a parallelogram. If $n=2$, then its area is $|u_1v_2-u_2v_1|$. If $n=3$, then its area is $\|u\times v\|$. Is there ...
3
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3answers
118 views

Number of matrices with zero determinant

I want to count matrices over finite field with zero determinant. For example, the numbet of $2\times2$ matrices over $\Bbb{Z}_4$ with zero determinant is 88 by hand computations. On the other hand ...
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0answers
30 views

Prove determinant of a Matrix using induction?

So, let $A\in \mathbb{M}_n$. I know that (without prooving it yet :D) - $\det A=\left\{ \begin{matrix}0 & n \text{ is odd}\\ 2 & n \text{ is even} \end{matrix}\right.$ $A =\begin{pmatrix} ...
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0answers
39 views

Numerical methods for the computation of a square matrix determinant

I have found many different ways to compute the determinant of a square matrix we'll call A over the internet : using LU decomposition, using Cholesky decomposition on the dot product $transp(A) . A$, ...
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0answers
15 views

Combination of matrixes [duplicate]

If A is a $k\times k$ matrix,B is a $k\times l$ matrix and C is a $l\times l$ matrix prove that: $\det{\begin{bmatrix}A&B\\O&C\end{bmatrix}}=\det(A)\det(C)$ O is the matrix that all it's ...
4
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2answers
57 views

On determinants of a certain class of matrices

$\newcommand{\M}{\mathcal{M}}$Recently I encountered a certain class of matrices whose determinants behave in an interesting manner. Define $\M(n,k)$ for positive integers $n,k$ with $k\leq n$ to be ...
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2answers
80 views

Proving that determinant is zero

Let $A,B \in M_3(\mathbb{C})$ such that $(AB)^2 = A^2B^2$ and $(BA)^2 = B^2A^2$. Prove that $\det(AB-BA) = 0$.
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1answer
37 views

Show the following matrix has determinant = 0

I just faced this problem where i am asked to show this matrix has determinant = 0 and I got stuck and can't find a way out of this...would really appreciate if someone could help $$ \begin{pmatrix} \...
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1answer
27 views

determinant involving unitary hermitian matrices

Let $S$ be the set of complex $N\times N$ matrices that are both unitary and hermitian. I have observed the following fact. For any pair of matrices $A$, $B$ from $S$, the value of $\det(A+iB)$ is ...
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0answers
36 views

$(-1)^{i+j}$ factor in the cofactor formula

Need the explanation of why there is a $(-1)^{i+j}$ factor in the cofactor formula. I thought that it might be an effect of the row swapping, which in turn affect the determinant sign, but I can't ...
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0answers
15 views

Determinant = product of the difference of 2 rows [duplicate]

Assume $x_1, . . . , x_n$ are numbers. Show that $$det\begin{pmatrix} 1 & x_1 & ... & x_1^{n-1} \\ 1 & x_2 & ... & x_2^{n-1} \\ & & ...\\ 1 & x_n & ... & ...
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0answers
18 views

Using LU Decomposition to find determinants

I've been trying to find advantages and disadvantages to using LU factorisation with pivoting to compute determinants. There's a lot of information on its usefulness in regards to solving systems of ...
1
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1answer
21 views

Question about the determinant of an upper triangular matrix

I was watching an online lecture on the properties of determinants and at 27:00 in this lecture: https://www.youtube.com/watch?v=srxexLishgY&t=4s, the professor did a step which I don't understand....
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1answer
28 views

How to prove the Cofactor Expansion Theorem for Determinant of a Matrix?

I understand that the definition of a determinant of a matrix implies that you can expand over the first row over the matrix, but how does that itself imply that you can expand over any row the ...