What is Determinant: Definition and 504 Discussions
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible, and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).
The determinant of a matrix A is denoted det(A), det A, or A.
In the case of a 2 × 2 matrix the determinant can be defined as

A

=

a
b
c
d

=
a
d
−
b
c
.
{\displaystyle {\begin{aligned}A={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=adbc.\end{aligned}}}
Similarly, for a 3 × 3 matrix A, its determinant is

A

=

a
b
c
d
e
f
g
h
i

=
a

e
f
h
i

−
b

d
f
g
i

+
c

d
e
g
h

=
a
e
i
+
b
f
g
+
c
d
h
−
c
e
g
−
b
d
i
−
a
f
h
.
{\displaystyle {\begin{aligned}A={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdhcegbdiafh.\end{aligned}}}
Each determinant of a 2 × 2 matrix in this equation is called a minor of the matrix A. This procedure can be extended to give a recursive definition for the determinant of an n × n matrix, known as Laplace expansion.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed ndimensional volume of a ndimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
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...
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# nonautomaticallyzero entries. By "nonautomaticallyzero" 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...
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
$$16((ln x)^2...
I need to find the values of ##\Omega## where ##(\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(\Omega^2 + i\gamma\Omega + \frac{2k}{3m})  (i\gamma\Omega)(i\gamma\Omega) = 0##
I get ##\Omega^4 2i\gamma \Omega^3  \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0##
I...
I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
In what cases it is better to call a thing "modulus" and in what cases "determinant"? In my algebra "determinant" is not a norm, discontinuous, positive for nonzero elements, not abiding triangle inequality. Should I better call it "modulus"?
I am [working][1] on the algebra of "divergencies", that is, infinite integrals, series and germs.
So, I decided to construct something similar to determinant of a matrix of these entities.
$$\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> $$
where ##V_\theta = V(r,t)##.
The Del operator in ##\{r,\theta,z\}$ is: $\vec{\nabla} = \left< \frac{\partial}{\partial r}, \frac{1}{r}\frac{\partial}{\partial \theta}, \frac{\partial}{\partial z}...
I think you all can see that ##a_{(i+1,j+1)} = a_{i,j} + a_{i+1,j} + a_{i,j+1}##
Now the determinant always give me problem. I have and idea to reduce this matrix by Chio to a 2x2 matrix and find the determinant of this 2x2.
Put i was not able to see any pattern to find what how the 2x2 matrix...
Hi, I have been having some trouble in finding the determinant of matrix A in this Q
Which relevant determinant property should I make use of to help me find the determinant of matrix A and maybe matrix B also
This is what I have tried for matrix A so far but it's not much help really
Any...
This is the question. The following is the solutions I found:
I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...
question:
My first attempt:
my second attempt:
So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...
So in particular, how could the determinant of some general "operator" like
$$ \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_{n1}\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...
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.
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...
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}$$Answer (red) via WAok I did not do any operations on this
Since by observation the 4th column can become all zero'showever didn't see anything in the book to support...
Homework Statement
Find the value of the solid's volume given by the ecuation 3x+4y+2z=10 as ceiling,and the cilindric surfaces
2x^2=y
x^2=3*y
4y^2=x
y^2=3x
and the xy plane as floor.The Attempt at a Solution
I know that we have to give the ecuation this form:
∫∫z(x,y)dxdy= Volume
So, in fact...
1. The problem statement:
Find out the maximum determinant of a matrix nxn which have just 1 and 1 elements.
2. The attempt at a solution:
I have tried for 2x2 and 3x3 matrices and so generalizing for nxn matrices. But I can’t figure out any pattern or something like that. Also, I barely know...
Hey! :o
For $n\in \mathbb{N}$ let $A_n$ be the real $n\times n$matrix with the elements \begin{equation*}a_{ij}=\begin{cases}i , &\text{ if } i=j1 \\ 1, & \text{ if } i=j \\ j, & \text{ if } i=j+1 \\ 0 , & \text{ otherwise } \end{cases}\end{equation*}
For $n=1, 2, 3$ we get the matrices ...
Homework Statement
I am trying to find the value of a determinant,
Homework Equations
See the notes given in my Textbook,
The Attempt at a Solution
I applied this method to find the value of a determiannt,
See it here,
Why is my result wrong?
I will be thankful for any help!
Find the cofactors of the elements in the second row of every determinant:
$$\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 (fourmomentum, spin, etc). Now I know this ends up calculating exactly like a 4x4 determinant, I'm just not quite...
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
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+ac=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...