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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6
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1answer
170 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 $A$. ...
3
votes
1answer
59 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
votes
3answers
108 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
150 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 ...
8
votes
4answers
753 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
547 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
126 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
votes
0answers
72 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
46 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 ...
7
votes
2answers
193 views

Determinant of exact sequence

Let $0 \to A \to B \to C \to 0$ be an exact sequence of vector spaces. I want to show that I have a canonical isomorphism $$\det(B)= \det(A) \otimes \det(C).$$ Here, "det" refers to the $n$-th ...
1
vote
3answers
105 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
vote
1answer
27 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} ...
0
votes
1answer
107 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
144 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
74 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
83 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
votes
1answer
64 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
votes
1answer
60 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
votes
1answer
38 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 $42$ ...
1
vote
1answer
103 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
vote
1answer
55 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 ...
2
votes
0answers
66 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
444 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
votes
1answer
32 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
votes
0answers
55 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
votes
1answer
20 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
votes
1answer
48 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
45 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 & ...
0
votes
1answer
101 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
votes
1answer
278 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
vote
1answer
106 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
130 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
vote
1answer
72 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
vote
0answers
38 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
vote
1answer
53 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
41 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 $0$. ...
2
votes
1answer
196 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
vote
1answer
131 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
vote
1answer
112 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
votes
1answer
42 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 & ...
1
vote
1answer
79 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
62 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
2k 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\\ ...
3
votes
2answers
87 views

Every skew-symmetric matrix has a non-negative determinant

Let $A$ be a skew-symmetric $n\times n$-matrix over the real numbers. Show that $\det A$ is nonnegative. I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ ...
1
vote
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
vote
1answer
62 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 ...
0
votes
4answers
135 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
votes
1answer
77 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
votes
2answers
88 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.
0
votes
3answers
55 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 ...