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
16 views

Changing order of determinant and limits

Given a Matrix X and another Matrix $M=e^X$ and $\det M =1$, I want to show that the Trace of X vanishes, e. g. $\mathrm{Tr} (X) =0$. I think that one can write it as following: \begin{equation} ...
0
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1answer
78 views

Matrix determinant lemma with adjugate matrix

I would like a proof of the following result, given on wikipedia. For all square matrices $\mathbf{A}$ and column vectors $\mathbf{u}$ and $\mathbf{v}$ over some field $\mathbb{F}$, $$ ...
3
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1answer
39 views

The determinant of any $3\times 3$ matrix of rank $2$ is $0$, and generalization

Here i want to show The determinant of of any $3\times 3$ matrix of rank $2$ is $0$. Can anyone give me a hint or proof for this? Further it is generalized to: for any $n\times n$ matrix of rank ...
1
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1answer
50 views

The product of $100\times 2$ and $2\times 100$ matrices must have zero determinant

Let A be a 100 × 2 matrix and let B be a 2 × 100 matrix. Then C = AB is a 100 × 100 matrix. Explain why $\det(C) = 0$. My initial thoughts on this question is that if $\det(C)=0$, then it cannot be ...
2
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1answer
37 views

Find $\det(C_n)$ where $c_{ij} = 1$ unless $i-j=\pm 1$

Find $\det(C_n)$ where $c_{ij} = 1$ unless $i-j=\pm 1$. The original problem from the quiz, Let $C_n$ be the $n$ by $n$ matrix whose entries are all ones, except for zeros directly below and ...
0
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0answers
27 views

Cauchy Determinant with Absolute Values

This is perhaps a straightforward question but I'm a little confused. An $n\times n$ Cauchy matrix $A$ is a matrix with entries $$a_{i,j}=\frac{1}{x_i-y_j}$$ for $1\le i,j\le n$, where $x_i$ and $y_j$ ...
1
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1answer
17 views

Density of $(\frac{Y_{i}}{\sqrt{\sum _{i}^{n}Y^{2}_{i}} })_{i}^{n}$ for iid $Y_{i}\in N(0,1)$ (Computing a determinant)

The density of $\left(\frac{Y_{i}}{\sqrt{\sum _{i}^{n}Y^{2}_{i}} }\right)_{i}^{n}$ for iid $Y_{i}\in N(0,1)$ can be computed by change of density formula. The answer will be rotationally symmetric. ...
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0answers
10 views

Comparing the orders of complexity.

I do have to evaluate a property defined as $$w = \sqrt{\det\left(\textbf{J}\,\textbf{J}^\text{T}\right)} = \prod_{i = 1}^n \sigma_i,$$ where $\textbf{J} \in R^{3 x n}$ and $n>3$. Using the ...
3
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2answers
61 views

Making a determinant easier to solve to find characteristic polynomial

I have to find a characteristic polynomial for the follwoing matrix: $$A=\left( \begin{matrix}0 & -1 & 1\\ -1 & 0 & 2 \\ 1 & 2 & 0 \end{matrix}\right)$$ My goal to find the ...
-3
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1answer
42 views

The determinant of permutation matrix that reverses the order of coordinates [closed]

How to find the determinant of $ n \times n $ permutation matrix of the following form? \begin{pmatrix} 0 & 0 & \cdots & 1 \\ 0 & 0 & 1 & 0 \\ \vdots & \vdots & ...
7
votes
1answer
97 views

The Diagonal Elements Of A Special Symmetric Matrix

A $n \times n$ matrix $M$ is a symmetric matrix,where $n$ is odd($i.e.n=2k+1,k\in \mathbb{Z}^{+}\cup{\{0\}}$). Every row of $M$ is a permutation of $\{1,2,\cdots,n\}$. Show that the diagonal ...
0
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0answers
11 views

Totally uni-modular matrix

I encountered the following matrix and am wondering whether it is totally uni-modular or not: $$\begin{bmatrix} A_{n\times m} & 0_{n\times m}\\ I_{n\times m} & I_{n\times m}\\ ...
3
votes
1answer
37 views

Proving additivity of determinant in each argument

I'm reading through Treil's Linear Algebra Done Wrong and have a question about proving a remark he leaves as an exercise on p.77. Here it is in my own words: Consider a function on $n$ vectors ...
7
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1answer
132 views

Area of triangle in determinant form

Area of triangle with vertex $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is given by : $$\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1 \end{vmatrix}$$ In this ...
4
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1answer
22 views

$a_{12},a_{23}$ and $a_{31}$ are the positive roots of the equation $x^3-6x^2+px-8=0,p\in R$ then find the value of $\det(A)$

Let $A=[a_{ij}]_{3\times 3}$ be a matrix.If $A+A^T= \begin{bmatrix} 6 & 4 & 4\\ a_{21}+a_{12} & 10 & a_{23}+a_{32}\\ a_{31}+a_{13} &4&8 \end{bmatrix}$ where ...
1
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1answer
23 views

Definition of determinant of a derivative.

Can someone please help me with the following definition: $B$ is a bounded open set in $R^n$ and $g$ : $\bar{B} \rightarrow R^n$ is $C^1$ . We say $a$ is a regular value of $g$ if ...
1
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2answers
76 views

if $A^2 = A$ then $|A|=0$ or $|A| =1$

More a verification of work then anything else, I am trying to prove the above statement. Intuitively I feel that if $A^2 = A$ then $\det(A) = \det(A^2)$. From here I know the property of $\det(AB) = ...
0
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1answer
92 views

Finding Eigenvalues of a 3x3 Matrix INVOLVING LAMBDA

Need some help with determinants involving eigen's. I understand the steps used in the process below, but I don't understand how my teacher knew that he had to do those steps to get a nice 0 row with ...
0
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3answers
158 views

How to find the determinant of a 5 by 5 matrix

How can find the value (determinant) for this matrix? I know in matrix $3\times 3$ $$A= 1(5\cdot 9-8\cdot 6)-2 (4\cdot 9-7\cdot 6)+3(4\cdot 8-7\cdot 5) $$ but how to work with matrix $5\times ...
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3answers
50 views

Use determinants to find which real values of c make each of the following matrices invertible

So i was given this question Use determinants to find which real values of c make each of the following matrices invertible $ \left[ {\begin{array}{cc} 0 & c & -c \\ -1 & 2& 1 \\ c ...
0
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1answer
93 views

Find det A if A is 3 × 3 and det(2A) = 6.

So i was given this question. Find det A if A is 3 × 3 and det(2A) = 6. Trying to solve this i tried to use the fact that If A is an n × n matrix, then $det(uA)$ = $u^ndetA$ for any number u. So ...
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3answers
82 views

Questions concerning $ \det ({}^A_{C\,}{}^B_D) = \det ({}^D_{B\,}{}^C_A)$

Let $A, B, C, D $ be $n \times n $ matrices. Using Schur complements I have found that $$ \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} A & 0 \\ 0 & I \end{pmatrix} ...
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0answers
13 views

Values of x for which S is not a basis of V3

Good morning, I have this problem to solve: In V3, given system S: (1,x,-1), (x,0,3), (1,3,-1), where x is a real number. Find all the values of x for which S is not a basis of V3. Below show the ...
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1answer
24 views

Let $\left\{\Delta_1,\Delta_2,…,\Delta_n\right\}$ be the set of all determinants of order 3 that can be made with the distinct real numbers

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

Three lines are concurrent (or parallel) $\iff$ the determinant of its coordinates vanishes.

I'm trying to prove the concurrency condition for three lines lying on a plane. This condition says that: Let \begin{cases} ax + by + cz=0 \\ a'x – b'y + c'z=0 \\ a''x + b''y + c''z=0 ...
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1answer
27 views

If $C=A-A^T$ and $a_{13}=1,a_{23}=-5,a_{21}=15$,then find the value of det(adj $A$)+det(adj$C$)

Let $A$ be a $3\times 3$ matrix given by $A=[a_{ij}]$ and $B$ be a column vector such that $B^TAB$ is a null matrix for every column vector $B.$If $C=A-A^T$ and $a_{13}=1,a_{23}=-5,a_{21}=15$,then ...
2
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1answer
31 views

Derivative of Determinant of Particle Trajectory

I have trouble with understanding vector calculus notation in a PDE book, much less matrix analysis. There is a proof in Majda's book "Vorticity and Incompressible Flow" very early on that I am not ...
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0answers
22 views

The determinant of the jacobi matrix of $x\mapsto \frac{x}{|x|^2}$

I want to calculate the determinant of the jacobian of the map $\phi$: $\mathbb R^n\rightarrow \mathbb R^n$, $x\mapsto \frac{x}{|x|^2}$ Since this maps doesnt seems to be linear I dont know how to ...
2
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1answer
53 views

Determinant form of quadratic equation, 3 variables, second order (nomogram)

I am looking for a determinant for a second order equation so that I can build a nomogram. The equation is simply: $$ x^{2} +2 a x-c = 0 $$ It can also be written in another format (which is more ...
2
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0answers
41 views

Three points of an affine space are collinear $\iff \det(A)=0$, with $A$ the matrix of the barycentric coordinates.

I'm doing this exercise: Let $3$ different points of an affine plane, with barycentric coordinates $X=(x_0,x_1,x_2), Y=(y_0,y_1,y_2), Z=(z_0,z_1,z_2)$ respect to a fixed reference frame. Prove ...
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4answers
86 views

Shorter way of evaluating the determinant of a $3\times 3$ Matrix

Is there any short way of evaluating this?-I try to do it, but i end up making mistakes somewhere. The answer is supposed to be(I'm sorry I don't know how to resize the image): But I never get ...
0
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1answer
48 views

Why determinant is defined in different way,when we have already simple definition?

I am reading the topic determinant from the book hoffman & kunze.they have defined determinant as follows: Let $K$ be a commutative ring with identity, and let $n$ be a, positive integer.Suppose ...
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1answer
46 views

How to solve $n^{th}$ determinant in complex domain using recurrence relations?

If $D_n=pD_{n-1}+rD_{n-2},n\ge 3$ is recurrence relation with constant coefficients, If $r=0\Rightarrow D_n=p^{n-1}D_1$ If $r\neq 0$ then solve equation $x^2-px-r=0$ If $$\Delta>0\Rightarrow ...
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0answers
59 views

How does the determinant change with respect to a base change?

Problem Suppose $k$ is a (commutative) field, and $A$ is a finite (dimensional) commutative unitary $k$-algebra. $M=A^n$ is a free $A$-module, and therefore can be seen as a finite-dimensional ...
8
votes
1answer
163 views

Validity of this Determinant property.

The property states that, "If to each element of any row (or column) of a matrix, product of a scalar and a corresponding element of any other row (or column) is added, the determinant of the new ...
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2answers
37 views

Calculating the determinant of a matrices

Show that $ det \left[ {\begin{array}{cc} 0 & 1 & 1 & 1\\ 1 & 0& x & x\\ 1 & x & 0 & x\\ 1 & x & x & 0 \end{array} } \right] $ = $-3x^2$ I have ...
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1answer
33 views

solving the determinants of matrices

So i was given this question Evaluate by first adding all other rows to the first row My solution: $ det \left[ {\begin{array}{cc} x-1 & 2 & 3 \\ 2 & -3& x-2 \\ -2 & x & -2 ...
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1answer
43 views

Find det A and det B

So i was given the question If a is $3 x 3$ and $det (2A^{-1})$ $=$ $-4$ $=$ $det(A^3(B^{-1})^T)$, find det A and det B. I'm completely confused how to go about this. I could not find a similar ...
0
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2answers
37 views

Determining the determinant of a matrix

So i was given this question Find det A if A is $3 × 3$ and $det(2A) = 6$. Under what conditions is det(−A) = det A? I'm used to dealing with questions that give a matrix to solve, but this question ...
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0answers
37 views

Without determinants, show a set in $L^n \subset K^n$ is linearly independent over $K$ if it is over $L$.

Let $L$ and $K$ be fields with $L \subset K$. Let $v_1,\ldots,v_r \in L^n$ be column vectors, linearly independent over $L$. Of course, we can also consider the vectors to sit in $K^n \supset L^n$. ...
0
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1answer
43 views

Matrix question regarding determinant, whether singular or not

Let $A$ be a nonzero square matrix such that $$A + A^2 + A^3 =0$$ Must $A$ be singular? If your answer is affirmative, give a proof, give a counterexample otherwise. I have ...
1
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1answer
32 views

Find the Rank and Signature of a Billinear Form

Let $V \in M_{2 \times 2} ^C$ be the set of all the herimitian of order 2. V is a linear space over $ \Bbb R$. Check that $q(A) =2det(A) $ is a Square Billinear Form. In addition, Find ...
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1answer
59 views

Relation of $det(AA^T)$ and $det(A^TA)$ with Range and Nullspaces of a Matrix [duplicate]

While I was studying for my exam, I found this question about range and nullspaces of a matrix, which seemed easy at the first glance, but I couldn't find a way to prove it. Could you please take a ...
0
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2answers
47 views

How to find the eigenvalues of this matrix, which has zeros on the diagonal and 1's elsewhere, without computing the characteristic polynomial,

I don't think that computing the characteristic polynomial is the way to proceed for this problem. Also, the size of this matrix C is just given as "nxn". So, instead, I can look at the matrix I+C. ...
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0answers
60 views

Find the value of this 3 by 3 determinant

I've tried to solve it with minors, but it doesn't look right. Could you solve it using determinant's properties?
0
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1answer
25 views

Find the value of the determinant.

I know it equals 0 beause I solved it using minors, but I should solve it using determinants' properties. I have just detarted the determinant into 2, but I can't do nothing else.
0
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4answers
53 views

Determinant of AB and BA, where A and B are $5\times 7$ and $7\times 5$ matrices over the real numbers.

I was doing a multiple-choice exercise where the only option left is $$\det(BA)=0.$$ Could someone explain why this is true?
0
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5answers
100 views

Prove that this determinant is $0$

$$ \begin{vmatrix} \cos 2x & \cos 2y & \cos 2u \\ \sin^2 x & \sin^2 y & \sin^2 u \\ 1 & 1 & 1 \end{vmatrix} = 0 $$ This is the conclusion where I got from another excercise ...
1
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0answers
25 views

Algebraic proof that $\det AB = \det A \cdot \det B$ using Leibniz formula for determinants [duplicate]

The Leibniz formula for determinants allows us to express an $n \times n$ matrix determinant as a sum over permutations in $S_n$: $$\det A = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \cdot ...
0
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0answers
30 views

Prove the identity of matrices

How should I solute it? Use the determinant properties.