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
29 views

Properties of Determinants in True or False Questions

These are some good practice problems for anyone searching on the Web for determinants problems. There is one or two questions that I am not getting right according to the system. Could you help me ...
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1answer
39 views

Switching rows of matrices and its effect on the value of the determinant.

I think there is a mistake here for the second determinant. When you switch rows twice, I believe you get the same determinant as the initial matrix. So the answer should be 3, not -3... Please ...
1
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0answers
28 views

Determinant from matrix of logarithms

Is there a way to get the determinant $\text{Det}(M)$ of a matrix $M$ from the matrix of its logarithms, i.e. $\Bigg( \begin{smallmatrix} \log(M_{00}) & \log(M_{01}) & \ldots \\ \log(M_{10}) ...
1
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1answer
48 views

Given three vectors involving trigonometric functions, how many $\theta$ satisfy a particular box product relation?

If $$\vec a =(1+\sin \theta )\hat i+\cos \theta \hat{ j}+\sin2\theta\hat k\\ \vec b =(\sin( \theta +2\pi/3))\hat i+\cos ( \theta +2\pi/3) \hat{ j}+\sin( 2\theta +4\pi/3)\hat k\\ \vec c =(\sin ( \theta ...
2
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1answer
36 views

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ smooth, $ g(x,y)= x^3 + y^3$ and $g \circ f \equiv 0$, then $\det Df \equiv 0$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function and $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $(x,y) \mapsto x^3 + y^3$. Assume that $g \circ f$ is identically $0$. ...
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0answers
37 views

Applications of Infinitary Matrices in Set Theory

Matrices have a natural generalization to infinitary context. There are few known applications of such matrices in set theory. For example one may use Ulam matrices to show that real-valued measurable ...
2
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1answer
45 views

How to find eigenvalues of this 3x3 Jacobian Matrix

I am having to learn how to do jacobian matrices, determinants, and finding eigenvalues on my own and I cannot seem to find reasonable eigenvalues for this jacobian matrix. When I try to solve it I ...
1
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1answer
29 views

Determinant equivalent of curl

$$\nabla \times V= \hat{e_x}\space(\frac{\partial}{\partial{y}} V_z-\frac{\partial}{\partial{z}} V_y)+\hat{e_y}\space(\frac{\partial}{\partial{z}} V_x-\frac{\partial}{\partial{x}} ...
1
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1answer
50 views

Interperate Jacobian Determinant - Stability of Equilibriums

In my SIR model, I have the following Jacobian Matrix \begin{align*} J =\begin{bmatrix} -\alpha I & -\alpha S & \zeta & 0 \\ \alpha I & \alpha S - \beta - \rho & 0 & 0 \\ 0 ...
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1answer
37 views

What does this determinant mean?

I have the following Jacobian matrix for an equilibrium of an SIR model $$J=\left( \begin{array}{cccc} -\text{$\alpha $N} & 0 & \zeta & 0 \\ \text{$\alpha $N} & -\beta -\rho & ...
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1answer
39 views

Give a general formula in terms of $n$ for the determinant of the following matrix.

Let $M_n$ denote the $n$ x $n$ matrix over $\mathbb{R}$ of which the entry in the $i$-th row and the $j$-th column equals $1$ if $|i-j|\leq 1$ and $0$ otherwise. For example: $M_6=$ \begin{pmatrix} ...
0
votes
2answers
51 views

Find the eigenvalues of the following matrix

Consider $A =\left( \begin{array}{ccc} -1 & 2 & 2\\ 2 & 2 & -1\\ 2 & -1 & 2\\ \end{array} \right)$. Find the eigenvalues of $A$. So I know the characteristic polynomial is: ...
3
votes
5answers
342 views

Is the determinant of this matrix positive or negative?

$\left( \begin{array}{ccc} 1 & 1000 & 2 & 3 &4\\ 5 & 6 &7&1000 &8\\ 1000&9&8&7&6\\ 5 & 4&3&2&1000\\ 1&2&1000&3&4\\ ...
2
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1answer
35 views

Every skew-symmetric matrix has a non-negative determinant

I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ skew-symmetric matrix). So when $n$ is odd, we have: $\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)\Rightarrow ...
1
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1answer
44 views

Can -3 and 2 be eigenvalues of the following matrix?

Can $-3$ and $2$ be eigenvalues of and nxn matrix B such that $A = B^{2}+B-6I$ and A's determinant is $0$? So this is what I concluded: At first glance, it can be seen that the matrix $A$ can be ...
1
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1answer
47 views

Does the determinant of a complex-valued matrix have a geometric interpretation?

The determinant of a real-valued matrix can be seen as the volume of the parallelotope with the column vectors as the sides. Is there an analogous interpretation for complex-valued matrix ...
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4answers
59 views

Prove determinant is zero

If $M = \begin{vmatrix} 1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \\ \end{vmatrix}$ Show that M = 0 WITHOUT expanding the determinant. I ...
2
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1answer
46 views

Write the determinant as a polynomial expression in the elementary symmetric polynomials

How to write $\det\begin{bmatrix}x_1&x_2&x_3&x_4\\x_2&x_3&x_4&x_1\\x_3&x_4&x_1&x_2\\x_4&x_1&x_2&x_3 \end{bmatrix}$in terms of elementary symmetric ...
0
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2answers
75 views

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____? This is a fill-in-the-blank problem that I found in my paper, but I don't have this answer.
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0answers
42 views

Find Eigenvalues of Infinite Matrix

I have the matrix $M$ acting on $l^2(\mathbb{N;C})$ given by the components $$ M_{n,n'} = V_n\delta_{n,n'} + A\delta_{n,n'+1}+A^\ast\delta_{n,n'-1} $$ where $V_n$ is real and obeys a periodicity ...
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3answers
46 views

Evaluating a determinant for eigenvalues

I need to evaluate $$\left| {\matrix{ {3 - \lambda } & 1 & 1 \cr 2 & {4 - \lambda } & 2 \cr 1 & 1 & {3 - \lambda } \cr } } \right|$$ A direct computation ...
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2answers
41 views

Use row operation to find the determinant?

Use row operations to find the determinant: Can someone give me a full answer please? Also can anyone tell me if the sign of the determinant matters ? Row operations : Det ( e(A) ) = ...
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1answer
25 views

Determinant of the Kronecker product involving the identity

Let $A$ be a square matrix and $I$ the $k \times k$ identity matrix. Then the identity $$ \det(A \otimes I) = \det(A)^k,$$ holds as can be seen from a general result on the determinant of block ...
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1answer
14 views

The maximum value of r?

A point $A = (a,b)$ is defined such that it lies on the graph $y = x^2 +1$ A point $B = (c,d)$ is defined such that it lies WITHIN the area of $ (x+2)^2 + (y+2)^2 = r^2$ Let's define a matrix $M = ...
-1
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2answers
17 views

Finding determinant of following matrix

I need to find determinant of following matrix . I did it by simply doing $R_5$ - $R_1$ . and then evaluating the determinant .But its a lengthy process but answer came out.. But another thing i have ...
-1
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1answer
23 views

To find the determinant in this question

Given $A$ by $4×4$ non singular matrix and $B$ be matrix obtained from A by adding to its third row twice the first row .Then $det(2A^{-1}B)$ is $A:2$ $B:4$ $C:8$ $D:16$ I cannot think anything ...
2
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2answers
41 views

Need help with determinant question

Can pls someone help me to understand rom how they have gone from first row in top determinant to first row in second determinant
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3answers
47 views

A quick way to generate 3x3 matrices with determinant equal to 1?

Perhaps a formula involving the row number and column number of an element or just some parametric equations for each element. I know that I can just multiply two of these matrices together to get ...
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0answers
24 views

Find the values of x,y,z so that the 3 x 3 matrix is singular?

Find the values of x, y, z that the matrix is singular? With an explanation.
2
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3answers
94 views

A $4\times4$ determinant with entries $\pm1$ is divisible by 8

Let B be a 4 by 4 matrix all of whose entries are -1 or 1 show det(B) is divisible by 8 Anyone can guide me for this? Thanks!
0
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1answer
43 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
2
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0answers
22 views

What is the simplest way to solve determinant of a $n \times n$ matrix by upper and lower triangular matrices?

I know the basic rules to solve for the determinant of an $n \times n$ matrix using upper and lower triangular matrices, but what is the simplest way?
3
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2answers
47 views

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 - \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that ...
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2answers
79 views

Is there a general form for the determinant of this matrix?

This came up in trying to deal with small oscillations of an $N$-pendulum. I obviously want to calculate the characteristic polynomial in $\omega^2$ to see if I can deal with the equation even in ...
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0answers
17 views

Trace of the exterior powers of linear operators

Given linear operators $K_1,\ldots,K_m$ on a Hilbert space $\mathcal H$, what can we say about the trace of their exterior product $Tr \,(K_1\wedge \cdots \wedge K_m)$ ? More precisely: 1) If we ...
1
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0answers
22 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
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2answers
42 views

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$. I would first assume that $A$ has rank $3$. Then $A'A$ would be a $4\times 4$ matrix with rank $3$ and therefore it would have ...
0
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1answer
35 views

A combinatorial coefficient linked to exterior product

I am looking at the following sum $$ \sum c_1\wedge \cdots\wedge c_n $$ where the summation ranges over $c_1,\ldots,c_n$ such that each $c_i\in\{a,b\}$ and $a$ appears exactly $j$ times. Thus, using ...
2
votes
1answer
39 views

Functions of several variables and $Df$

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $g(x_1,...,x_n)=x_1^5+...+x_n^5$. Suppose $g\circ f\equiv 0$. Show that ...
2
votes
1answer
74 views

Invertibility of block matrices, with the property of being symmetric, positive definite, and of full rank:

If A and B are real matrices, with A being symmetric, B having at least as many columns as rows, and the matrix C defined as: $$ \begin{bmatrix} A & B^T \\ B & 0 \\ ...
0
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3answers
89 views

How do I find the determinant of a 4x4 matrix when the diagonal is made up of variables? [closed]

Evaluate: $\det(A)$, where $A= \begin{bmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a\end{bmatrix}$
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2answers
52 views

Determinants of $3\times3$ matrices with full rank

I have two $3\times3$ matrices $A$ and $B$ where $$A = [c_1 : c_2 : c_3]$$ $$B = [c_1 : c_1 + c_2 : c_1+c_2+c_3]$$ where $c_i$ is the $i^{th}$ column of $A$. Given that $|A| = 1$, I am to find the ...
0
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1answer
43 views

$2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My guess ...
3
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3answers
268 views

For which $x$ is the determinant 0?

For which values of $x \in \mathbb{R}$ does the determinant of the matrix $$ M = \begin{pmatrix} x & 0 & 1 & 2 \\ 2 & x & 0 & 1 \\ 1 & 2 & x & 0 \\ 0 & 1 ...
0
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0answers
44 views

matrix determinant changes when doing row operation, so weird O_o

To calculate the determinant of a matrix, you can subtract a row by another, and the determinant will not change. However, in the following matrix, the determinant is -2. \begin{bmatrix} 1 ...
2
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0answers
53 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
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1answer
61 views

Cramers Rule. The why and how.

Can someone explain how Cramer's rule works. I understand the mechanics of it, and it's fairly straightforward to show algebraically that it's equivalent to GJ and substitution, but what's happening ...
0
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2answers
64 views

Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...
-1
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4answers
62 views

If $A$ is a $3 \times 3$ matrix and $\det(A) = 4$, then compute $\det(((-9A)^4)^T)$. [closed]

Given a $3\times3$ matrix $A$ $$A= \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$ and $\det(A)=4$ Calculate $\det(((-9)\cdot A)^4)^T$.
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2answers
24 views

If rank of $(m+1)\times n$ matrix is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant.

I can't understand this : If I have a $(m+1)\times n$ matrix and if its rank is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant. How is it so?... kindly help.