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
35 views

How to simplify $\det(M)=\det(A^T A)$ for rectangular $A=BC$, with square diagonal $B$ and rectangular $C$ with orthonormal columns?

Assume a real, square, symmetric, invertible $n \times n$ matrix $M$ and a real, rectangular $m \times n$ matrix $A$ such that $m \geq n$ and $M = A^T A$. Also assume that $A = B C$, where $B$ is ...
2
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2answers
47 views

Determinant of a 4x4 matrix with trigonometric functions

I am stuck with my homework from math. I should calcutate the determinant of a matrix: $$\begin{bmatrix} sin(x) & \sin(2x) & \cos(x) & \cos(2x)\\ cos(x) & 2\cos(2x) & ...
8
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1answer
59 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
2
votes
1answer
33 views

Determinantal inequality for block matrices: if $A=(B,C)$ is a square matrix, then $|A|^2\le |B^TB|\cdot |C^TC|$

Suppose $A=(B,C)$ is a $n\times n$ matrix, $B$ is a $n\times s$ matrix, $C$ is a $n\times (n-s)$ matrix. Show that $|A|^2\leq |B^TB|\cdot |C^TC|$. If $A$ is singular, then it is obvious. If $A$ is ...
0
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0answers
23 views

What is the proof of this? (Matrices, Pivot)

I have a matrix : $A$ I pivoted $A$ with a pivot element $(p)$ and I get this matrix: $B$ What is the proof of this equation? $|A|$ = $\frac{1}{p}. |B|$
0
votes
3answers
79 views

Why does the determinant $D$, have to be $0$ for equation to have a solution?

Suppose $2\times2$ equation: $$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$ We can make determinants: ...
7
votes
1answer
76 views

Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
7
votes
4answers
661 views

A faster way of calculating this determinant?

I'm doing a problem involving Cramer's rule, and one of the determinants I have to work with is as follows: \begin{vmatrix} 1&1&1\\ a&b&c\\ a^3&b^3&c^3 \end{vmatrix} So I ...
0
votes
0answers
37 views

How to show a set of vectors does not span a vector space?

Let's say I am given a $4\times4$ matrix and I am to determine whether the columns of that matrix span $\mathbb R^4$. Please tell me if I'm correct: One way to determine that is to calculate the ...
0
votes
1answer
17 views

Algorithm for the Hill cipher (finding the inverse of the determinant of a $2 \times 2$ matrix modulo $26$)

I have a good understanding of how to do the Hill cipher on paper but putting it into program form is somewhat of a problem. Finding the the determinant is the thing I'm having problem with. On ...
3
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0answers
37 views

Finding $n$ scalars such that $\det{(cI-A)}=0$ without eigenvalues

My problem is this Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$ I know that the determinant is ...
2
votes
2answers
36 views

Linear Algebra: Properties of the Determinant

On a recent exam, I was given the following problem: Suppose that $\det(A) = -3$, $\det(A + I) = 2$, and $\det(A + 2I) = 5$. What is $\det(A^4 + 3A^3 + 2A^2)$? I just don't see how the ...
1
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3answers
41 views

Solving variables in a matrix for a specific determinant

The matrix is as follows: $$ A = \begin{pmatrix} 0 & x & 1 & 2 \\ x & 1 & 1 & x \\ 1 & x & x & 1 \\ 1 & x & 1 & x \end{pmatrix} $$ What I want to do ...
1
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1answer
19 views

Help with inverse matrix problem? (Specific problem in description)

\begin{equation} \text{If} \begin{vmatrix}A\end{vmatrix} \text{=}\frac{1}{24} \text{, solve } \begin{vmatrix} \begin{pmatrix}\frac{1}{3}A\end{pmatrix}^{-1} - 120 \text{ }A^* \end{vmatrix} ...
1
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1answer
63 views

Proving determinant equality \begin{equation}\det{((A+B)^2)} = [\det(A+B)]^2\end{equation}

This is what we have to prove or disprove: \begin{equation}\det{((A+B)^2)} = [\det(A+B)]^2\end{equation} However, I really have no idea where to start - I tried plugging in two random sets of ...
4
votes
3answers
87 views

Help with resolving an n x n determinant?

I'm still a beginner, and would appreciate any tips regarding this. (Full solution appreciated, but hints more so!) This is the problem. \begin{equation}{D_n} = \begin{vmatrix} 1+{a_1} & 1 ...
0
votes
2answers
34 views

Square matrices as a product of elementary matrices,

I am trying to prove det(A) = det($A^T$), starting with the idea that every square matrix is the product of elementary matrices. Is this true, even for the non-invertible square matrices? So, I'd ...
4
votes
0answers
39 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
-4
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1answer
56 views

Why does the determinant of a $4 \times 4$ matrix contain $24$ products?

$(a)$ If $a_{11}=a_{22}=a_{33}=0$, how many of the $6$ terms in $det A$ will be zero? $(b)$ If $a_{11}=a_{22}=a_{33}=a_{44}=0$, how many of the $24$ products $a_{1j}a_{2k}a_{3l}a_{4m}$ are sure ...
0
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1answer
31 views

Determinant on 3x3 matrix and above

When finding the determinent of a matrix, what is the rationale behind multiplying the entry along the row we are deleting from times the cofactor expansion? Also how does doing cofactor expansion ...
0
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1answer
23 views

Linear Algebra - Determinant of linear transformation

So I'm working through sample questions and this came up. Any help would be greatly appreciated. Question Let $V$ be the vector space of all complex-valued polynomials $p(x)$ of degree at most $42$ ...
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1answer
24 views

Proving the determinant on the L.H.S = determinant on the R.H.S.

Prove the below determinants are equal without expanding them. \begin{vmatrix} {\alpha a_2} + {a_3}&{\beta a_3} + {a_1} & {\gamma a_1} + {a_2} \\ {\alpha b_2} + {b_3}&{\beta b_3} + ...
1
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1answer
49 views

What is the correct $\det(A^{-1})$

Ok so I think I know why this is incorrect, because of the following: $$\det\frac{1}{ad-bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\neq \frac{ad-bc}{ad-bc}$$ However, by adding a det ...
1
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0answers
44 views

Reference request on a sum-of-determinants identity

Suppose $X_1,X_2,X_3\in\mathbb R^{2\times1}$. Then $$ \det[ X_1,X_2] +\det[X_2,X_3] + \det[X_3,X_1] = \det[X_2-X_1,X_3-X_1]. $$ Where are this identity and higher-dimensional versions and their ...
4
votes
1answer
64 views

Det(AB)=0: what is the determinant of A and B

True or false. If the determinant of AB is zero, then the determinant of A is zero or the determinant of B is zero. I put true in my exam. After all det(A)det(B)=det(AB). Why was I wrong? The answer ...
0
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1answer
23 views

$A , B$ square matrices of size $n$ with real entries with $B$ invertible , the does $\exists c \in \mathbb R$ such that $\det (A+cB)=0$?

Let $A$ be a $n \times n$ matrix with real entries and $B$ is an invertible $n \times n$ matrix with real entries ; then does there exist $c \in \mathbb R$ such that $\det(A+cB)=0$ ?
0
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0answers
25 views

Hermitian Matrix Determinant

I have the following question to prove that $$\overline {( A)} = \overline{\det(A)}$$ The question does not state anything else, so I am not sure if A is Hermitian (this question is under the ...
0
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1answer
16 views

Cross product problem

someone could show me the error in the cross products? For $U=x\hat{i}+y\hat{j}+z\hat{k}$, $V=x'\hat{i}+y'\hat{j}+z'\hat{k}$ and $((.))$=modulus, we have $$U \times V=((U))((V))sin(U,V).n = ...
0
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1answer
18 views

Determinant and Discriminant as volume

As described here, the volume of the n-dim. parallepiped with $v_i$ in $\mathbb R^n$ as common edges from the originb is the abs. value of the determinant of the linear transformation taking the ...
0
votes
1answer
27 views

Determinant of a matrix of size n

I received a matrix for which I need to calculate its determinant. $$ A = \begin{pmatrix} 0 & 1 & 1 & \cdot & \cdot & \cdot & 1 \\ 1 & 0 & 1 & 1 & \cdot & ...
1
vote
1answer
30 views

IfA is an upper triangular n x n matrix, then det(A) is not equal to 0 . Why is this false?

I am studying linear algebra and the book just confused me in a way I can't explain. If A is an upper triangular n x n matrix, then det(A) is not equal to 0. The book says this is false. Can someone ...
0
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1answer
18 views

general idempotent matrix possible values of the determinant

If A is a general idempotent matrix, calculate the possible values of det (A) I caculated the det = o what other values can it equal?
1
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1answer
20 views

Determinant Algebra question; finding a determinant based on other matrices

! First part was very simple, second part though I've been wrking on it and am still confused. Okay now let me start by saying that I know all the rules for determinant algebra (I think). The thing ...
0
votes
1answer
26 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 ...
1
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1answer
38 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
24 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}) ...
0
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0answers
11 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
votes
1answer
34 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 $0$. ...
0
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0answers
36 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
27 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
27 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
42 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 ...
0
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1answer
31 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
37 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
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0answers
28 views

Matrix nxn Calculate

Calculate the determinant of the following nxn matrix as a function of n??
0
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2answers
48 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
324 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
votes
1answer
30 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
vote
1answer
42 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
vote
1answer
42 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 ...