Determinant Definition and 494 Threads

A matrix nxn with a parameter p is given and the question is what is the rank of that matrix in terms of p, the gaussian elimination is the standard process and i know how to do it. But i was wondering if the determinant of a matrix tells us if the matrix has independent columns thus telling us...

I have a doubt about this problem. (a) Show that a matrix ##\left(\begin{array}{ll}e & g \\ 0 & f\end{array}\right)## has determinant equal to the product of the elements on the leading diagonal. Can you generalize this idea to any ##n \times n## matrix? The first part is simple, it is just ef...

Let ##M## be a real ##n \times n## matrix. If ##M + M^T## is positive definite, show that $$\det\left(\frac{M + M^T}{2}\right) \le \det M$$

Hello, I need some advice because I just can't figure out how to solve the problem. I could try to make the determinant triangular by adding all the b together, but that doen't seem a good way of solving the problem. Is there any direction I should be thinking of? Thanks

Here is an example of the decomposition for a 2 x 2 matrix We have ##2^2=4## determinants, each with only #n=2# non-automatically-zero entries. By "non-automatically-zero" I just mean that they aren't zero by default. Of course, any of ##a,b,c##, or ##d## can be zero, but that depends on the...

program main ! use ! some library that defines the function to calculate the determinant of a given matrix implicit none real,dimension(2,2)::A real::det_val A(1,1)=1 A(2,2)=1 A(2,1)=0 A(1,2)=0 ! det_val=det(A) print *,det_val ! Should print 1. end program main

Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...

Let me first list the four axioms that a determinant function follows: 1. ## d (A_1, \cdots, t_kA_k, \cdots, A_n)=t_kd(A_1, \cdots A_k, \cdots, A_n)## for any ##A_k## and ##t_k## 2. ##d(A_1, \cdots A_k + C , \cdots A_n)= d(A_1, \cdots A_k, \cdots A_n) + d(A_1, \cdots C, \cdots A_n)## for any...

M. Blennow's book has problem 2.18: Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of...

By definition, ##\det A=\sum_{p_j\in P}\textrm{sgn}(p_j)\cdot a_{1j_1}\cdot\ldots\cdot a_{nj_n}##, where ##P## denotes the set of all permutations of the ordered sequence ##(1,\ldots,n)##. Denote the number of permutations needed to map the natural ordering to ##p_j## as ##N_j##. Now consider...

We know that the cofactor of determinant ##A##, is $$\frac{\partial A}{\partial a^{r}_{i}} = A^{i}_{r} = \frac{1}{2 !}\delta^{ijk}_{rst} a^{s}_{j} a^{t}_{k} = \frac{1}{2 !}e^{ijk} e_{rst} a^{s}_{j} a^{t}_{k}$$ By analogy, $$\frac{\partial Z}{\partial Z_{ij}} = \frac{1}{2 !}e^{ikl} e^{jmn}...

The way I approach it was, we're looking for det(H) where H = h(u, v) $$H = \begin{bmatrix} du/da & du/db \\ dv/da & dv/db \end{bmatrix} * \begin{bmatrix} da/dx & da/dy \\ db/dx & db/dy \end{bmatrix}$$ I just multiply those two matrices and then get the determinant. The answer is $$16((ln x)^2...

I need to find the values of ##\Omega## where ##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0## I get ##\Omega^4 -2i\gamma \Omega^3 - \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0## I...

I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...

In what cases it is better to call a thing "modulus" and in what cases "determinant"? In my algebra "determinant" is not a norm, discontinuous, positive for non-zero elements, not abiding triangle inequality. Should I better call it "modulus"?

I am [working][1] on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\det w=\exp(\operatorname{reg }\ln w)$$ which is analogous to how determinant of a matrix can be...

I have a vector in cylindrical Coordinates: $$\vec{V} = \left < 0 ,V_{\theta},0 \right> $$ where ##V_\theta = V(r,t)##. The Del operator in ##\{r,\theta,z\}$ is: $\vec{\nabla} = \left< \frac{\partial}{\partial r}, \frac{1}{r}\frac{\partial}{\partial \theta}, \frac{\partial}{\partial z}...

Find determinant of following matrix: ## A = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n-1} & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n-1} & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n,1} & 0 & \cdots & 0 & 0 \end{pmatrix} ## Note: I tried to solve this question...

I think you all can see that ##a_{(i+1,j+1)} = a_{i,j} + a_{i+1,j} + a_{i,j+1}## Now the determinant always give me problem. I have and idea to reduce this matrix by Chio to a 2x2 matrix and find the determinant of this 2x2. Put i was not able to see any pattern to find what how the 2x2 matrix...

$\tiny{311.3.2.16}$ Find the determinants where: $\left|\begin{array}{rrr}a&b&c\\ d&e&f\\5g&5h&5i\end{array}\right| =a\left|\begin{array}{rrr}e&f \\5h&5i\end{array}\right| -b\left|\begin{array}{rrr}d&f \\5g&5i\end{array}\right| +c\left|\begin{array}{rrr}d&e\\5g&5h\end{array}\right|=$ ok before...

Hi, I have been having some trouble in finding the determinant of matrix A in this Q Which relevant determinant property should I make use of to help me find the determinant of matrix A and maybe matrix B also This is what I have tried for matrix A so far but it's not much help really Any...

This is the question. The following is the solutions I found: I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...

question: My first attempt: my second attempt: So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...

So in particular, how could the determinant of some general "operator" like $$ \begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix} $$ with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary...

I don't have a clue as to how to go about proving (or verifying) the equation above. It would be very hard to take individual values of i,j and k and p,q and r for each side and evaluate ##3^6## times! More than that, I'd like a proof more than a verification. Any help would be welcome.

Hey! Let $\mathbb{K}$ be a field and let $1\leq n\in \mathbb{N}$. Let $a_0, \ldots , a_{n-1}\in \mathbb{K}$ and let $m_n\in M_n(\mathbb{K})$ be given by \begin{equation*}m_n:=\begin{pmatrix}0 & 0 & \ldots & 0 & -a_0 \\ 1 & \ddots & \ddots & \vdots & \vdots \\ 0 & \ddots & \ddots & 0 & \vdots...

Hey! :o Let $Ax=b$ be a system of linear equations, where the number of equations is by one larger than the number of unknown variables, so the matrix $A$ is of full column rank. Why can the test for combatibility of equations use the criterion of the determinant $|A \ b|$ ? (Wondering)

Hey! :o We have the matrix $A=\begin{pmatrix}a_1 & b_1 \\ a_2 & b_2\end{pmatrix}$. We consider the vectors $\vec{v}:=A\vec{e}_1$ and $\vec{w}:=A\vec{e}_2$. Justify geometrically, why the area of the parallelogram spanned by $\vec{v}$ and $\vec{w}$ is equal to $\det A$. Calculate the...

I am trying to show that given the following stochastic differential equation: ##\dot{x} = W(x(\tau))+\eta(\tau),## we have ##det|\frac{d\eta(\tau)}{dx(\tau')}| = exp^{\int_{0}^{T}d\tau \,Tr \ln([\frac{d}{d\tau}-W'(x(\tau))]\delta (\tau - \tau'))} = exp^{\frac{1}{2}\int_{0}^{T}d\tau...

Help me if what I have done so far can be simplified further.

I assumed that my calculation would be 3(-5^-1)(6) and I got the answer -18/5, however this is incorrect, I am unsure of where I am going wrong. I thought the determinant of a matrix is equal to the determinant of the transpose of the matrix so det(B)=6 would also be det(B^T)=6? Thank you.

On that sentence, cofactor of an element of a metric is derived. But I can not verify it. Here I attached the copy of the page.

In this paper ##J=\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial f_2(X_2)}{\partial X_2}\frac{\partial f_3(X_3)}{\partial X_3}## where ##f_2(X_2),f_1(X_1),f_3(X_3)## evolves with time. Now using this ##\dot J=\frac{d}{dt}(\frac{\partial f_1(X_1)}{\partial X_1}\frac{\partial...

I came across a line in this paper at page (2) at right side 2nd para where it is written ##d^3x=Jd^3X## where ##J## is the Jacobian and x and X are the positions of the fluid elements at time ##t_0## and ##t## respectively. Here what I have concluded that ##x_i=f(X_i)## where the functional...

where does 32 (red) come from ? nevermind looks its (-8)(4)=-32 but will probable have more ? on this example

Homework Statement The problem is to calculate the determinant of 3x3 Matrix by using elementary row operations. The matrix is: A = [1 0 1] [0 1 2] [1 1 0] Homework EquationsThe Attempt at a Solution By following the properties of determinants, I attempt to get a triangular matrix...

$$\left[\begin{array}{rrrrr} 1 &0 &2 &1\\ 1 &1 &0 &1\\ 1 &3 &4 &1\\ -1 &-3 &-4 &-1 \end{array}\right]=\color{red}{0}$$Answer (red) via W|Aok I did not do any operations on this Since by observation the 4th column can become all zero'showever didn't see anything in the book to support...

Compute the determinant of the following matrix $$\left| \begin{array}{cccc} 2 & 1 & 0 & 2 \\1 & 2 & 1 & 2 \\-1 & 1 & -3 & 2 \\1 & -1 & 1 & 0 \end{array} \right| \sim \left| \begin{array}{cccc} 2 & 1 & 0 & 2 \\1 & 2 & 1 & 2 \\-1 & 1 & -3 & 2 \\0 & 0 & -2 & 2 \end{array} \right| \sim \left|...

Homework Statement Find the value of the solid's volume given by the ecuation 3x+4y+2z=10 as ceiling,and the cilindric surfaces 2x^2=y x^2=3*y 4y^2=x y^2=3x and the xy plane as floor.The Attempt at a Solution I know that we have to give the ecuation this form: ∫∫z(x,y)dxdy= Volume So, in fact...

1. The problem statement: Find out the maximum determinant of a matrix nxn which have just 1 and -1 elements. 2. The attempt at a solution: I have tried for 2x2 and 3x3 matrices and so generalizing for nxn matrices. But I can’t figure out any pattern or something like that. Also, I barely know...

Hey! :o For $n\in \mathbb{N}$ let $A_n$ be the real $n\times n$-matrix with the elements \begin{equation*}a_{ij}=\begin{cases}i , &\text{ if } i=j-1 \\ 1, & \text{ if } i=j \\ -j, & \text{ if } i=j+1 \\ 0 , & \text{ otherwise } \end{cases}\end{equation*} For $n=1, 2, 3$ we get the matrices ...

Homework Statement I am trying to find the value of a determinant, Homework Equations See the notes given in my Textbook, The Attempt at a Solution I applied this method to find the value of a determiannt, See it here, Why is my result wrong? I will be thankful for any help!

Find the cofactors of the elements in the second row of every determinant: $$\begin{vmatrix}-2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \end{vmatrix}$$ I am going to guess that I need to look at each number in the second horizontal row to see what i and j are for finding the cofactors of the elements...

We can denote the jacobian of a vector map ##\pmb{g}(\pmb{x})## by ##\nabla \pmb{g}##, and we can denote its determinant by ##D \pmb{g}##. We were asked to prove that ##\sum_j \frac{\partial ~ {cof}(D \pmb{g})_{ij}}{\partial x_j} = 0## generally holds so long as the ##g_i## are suitably...

Hi all, I'm doing undergraduate research this summer, and a few times I've been told to calculate a term with the following form: ∈abcdpaqbkcsd, where p,q,k and s are four vectors (four-momentum, spin, etc). Now I know this ends up calculating exactly like a 4x4 determinant, I'm just not quite...

Homework Statement Given a matrix ##A = (a_{ij})##, we define matrix ##B = \begin{pmatrix} a_{11} & - a_{12} & a_{13} & \cdots \\ - a_{21} & a_{22} & -a_{23} & \cdots \\ a_{31} & - a_{32} & a_{33} & \cdots \\ \vdots & \vdots & \vdots & \vdots \end{pmatrix}##. Another way to define ##B## is...

Challenge Problem: Let $A$ be an $r \times r$ matrix with distinct eigenvalues $λ_1, . . . , λ_r$. For $n \ge 0$, let $a(n)$ be the trace of $A^n$. Let $H(n)$ be the $r \times r$ the Hankel matrix with $(i, j)$ entry $a(i + j + n - 2)$. Show that $ \displaystyle \lim_{n \to \infty} \lvert...

Homework Statement Q54. Homework Equations Cofactor expansion (along 1st column) The Attempt at a Solution

Homework Statement Refer given image. Homework Equations Expansion of determinant. w^2+w+1=0 where w is cube root of 1. The Attempt at a Solution Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...