What is Determinants: Definition and 169 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.
Hi,
In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment.
Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with...
Hi,
unfortunately, I have problems with the task c
I used the tip with the Laplace evolution theorem and rewrote the determinant to calculate ##W_n##. Then I simply formed the scalar product ##W_1 W_n## and here I get now no further.
How can we prove that
$$
g(A_1, \cdots A_n)= c g(I_1 \cdots I_n)$$?
From the those three axioms we can prove a property of g that if any of two vectors in domain exchange their respective places the sign of output of g will be changed.
Now, do we have to argue that any matrix can be changed...
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...
Hi,
I'd like to have a little insight about why the determinants of ℝ2x2 and ℝ3x3 matrices are computed that way.
I know how to calculate said determinants in both the cases and I also know what's the meaning behind it thanks to "3blue1brown"'s youtube channel, which states that they are a...
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...
In Loring W. Tu's book: "An Introduction to Manifolds" (Second Edition) ... Proposition 3.27 reads as follows:
The above proposition gives the wedge product of k linear functions as a determinant ...Walschap in his book: "Multivariable Calculus and Differential Geometry" gives the definition of...
I am reading Gerard Walschap's book: "Multivariable Calculus and Differential Geometry" and am focused on Chapter 1: Euclidean Space ... ...
I need help with an aspect of the proof of Theorem 1.3.1 ...
The start of Theorem 1.3.1 and its proof read as follows:
I tried to understand how/why...
nmh{898}
311 Determinants And Cofactor Expansion (3.1.1)
a. Compare the determinants using a cofactor expansion across the first row.
b. compute the determinant by a cofactor expansion down the second column.
$$\left
\begin{array}{rrr}
3&0& 4\\
2&3& 2\\
0&5&1\\
\end{array}
\right$$
ok I...
Dear all,
I was trying to prove that the area of a triangle is equal to the determinant consisting of the three points of the triangle. I got to the end, and something ain't working out. The signs are all wrong.
In the attached pictures I include my proof. Can you please tell me how can the...
Let $a$ and $b$ be real coefficients ($b \ne 0$), and let $(x^2+ax+b)^{1} = \sum_{k=0}^{\infty}c_kx^k$
for sufficiently small $x$.
Show, that the ratio of determinants:
$\begin{vmatrix} c_k & c_{k+1} \\ c_{k+1} & c_{k+2} \end{vmatrix} / \begin{vmatrix} c_{k+1} & c_{k+2} \\ c_{k+2} &...
Homework Statement
"Determine whether the function ##T:M_{2×2}(ℝ)→ℝ## defined by ##T(A)=det(A)## is a linear transformation.
Homework Equations
##det(A)=\sum_{i=1}^n a_{ij}C_{ij}##
The Attempt at a Solution
I'm assuming that it isn't a linear transformation because ##det(A+B)≠det(A)+det(B)##...
Why do most books on linear algebra have something like "Determinants are useless now".I have seen this in Strang, Friedberg and Axler's book.
Are determinants of no use in Maths ? which tool has taken its place in algebra ? And why this happened ?
Homework Statement
Let ##A## and ##B## be ##n \times n## matrices
1) Suppose ##A^2 = 0##. Prove that ##A## is not invertible.
2) Suppose ##AB=0##. Could ##A## be invertible.
3) If ##AB## is invertible, then ##A## and ##B## are invertible
Homework EquationsThe Attempt at a Solution
1) Suppose...
Let A and B be nxn matrices over an arbitary field such that AB = BA. Prove that if n is odd then A or B is not invertible.
This is rather easy when we use determinants. However, I am curious, how hard would it be to prove this without the use of determinants? What would be involved in such a...
Hello!
Please, help me to see my mistake  for quite a while I can't solve a very easy matrix.
I have to
find the inverse of the given matrix using their determinants and adjoints.
4 6 3
3 4 3
1 2 6
to find adjoint matrix I need to find cofactors 11, 12, etc till 33.
Cofactor11 =...
Homework Statement
Homework EquationsThe Attempt at a Solution
The answer in the solutions is given as : (2x+1)(x1)(1x), they did their matrix differently so that's how they got that answer. I used wolfram alpha to factorise my quadratic on the last line and it gave me alternative forms...
Hello I was learning about determinants and matrices. I learned the generalization of getting the determinant of an n by n matrix. I then applied this to vector space (i + j + k) via a cross product and noticed that you leave the i j and k in their own columns in the first row of the matrix...
I have been reviewing linear algebra for my FE exam, and I was thinking about cofactors. What are these strange things? It totally mystifies me that you can make a cofactor matrix from a matrix A (where the does alternating +/ come from??), transpose it, find the determinant (I still don't...
How does one create the single or doubly excited slater determinants in CI? What I mean is using HF, when I get the MO's, I can create the HF slater determinant which runs over electrons in rows and orbitals in columns. What is the form of single or doubly excited slater determinants? Sorry if...
In HS they just taught you the formula for it (the cofactor method) and a few other things, such as det(A) = 0 means A is singular.
I finally reached Ch 5 of MIT OCW Intro to Linear Algebra, and I was really hoping that seeing how determinants are derived from first principles would give me...
Consider a system of 2 identical fermions.
$$\psi_{k_1,k_2}(x_1,x_2,m_1, m_2) = \langle x_1\,x_2\,m_1\,m_2\mid \psi \rangle$$
According to what I have read we can construct a state with the right antisymmetry properties by
$$\psi_{k_1,k_2}(x_1,x_2,m_1, m_2) =...
Homework Statement
The moment of the couple is 600k (Nm). What is the angle A?
F = 100N located at (5,0)m and pointed in the positive x and positive y direction
F = 100N located at (0,4)m and pointed in the negative x and negative y direction
Homework Equations
M = rxF
M = DThe Attempt at a...
I'm taking a Differential Equations class and we're dealing with matrices and determinants. I've dealt with them before but I was always annoyed by the fact that I don't know what the heck is going on. So I know that matrices are a way to organise linear equations and make transformations...
Homework Statement
I'm a bit at a loss  I thought the last row with '1's would be useful, but it just gave me:
(b2c  bc2)  (a2c  ac2) + (a2b  ab2)
and
bc(b  c)  ac(a  c) + ab(a  b)
But then it is a dead end. I am probably doing something stupid again ...
Any help appreciated.
I just did this following exercise in my text
If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that
\int_C xdy  ydx = x_1y_2  x_2y_1
I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...
Hello, I have been studying matrices and determinants recently and do not understand why certain things are done the way they are. Like, why is matrix multiplication defined the way it is.
I find that there are not enough proofs. Is there any book/article/video that any of you recommend to...
I got three equations:
lcmbn=0
cl+man=0
blam+n=0
In my textbook, its written "eliminating l, m, n we get:"
$$
\begin{vmatrix}
1& c& b\\
c& 1& a\\
b& a& 1\\
\end{vmatrix}=0
$$
but if I take l, m, n as variables and since ##l=\frac{\Delta_1}{\Delta}## (Cramer's rule) and...
Homework Statement
Prove that if A is an n x n matrix with the property A3=A, then det(A)=1, det(A)=0, or det(A)=1
Homework EquationsThe Attempt at a Solution
At first I started with the property A3=A
I then applied the determinant to both sides.
From this point I don't really see any...
How we cannot apply row and column operation simultaneously on matrix when finding its inverse by elementary transformation but can apply it in determinant?
I think kernel and image gets disturbed in a matrix, though I don't know what it actually is.
Why not in determinant case?
Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...
Any1 can explain to me why det(C)=det(A)^(n1)
where A is nbyn matrix and C is the matrix of cofactors of A.
I have been thinking, any 1 can help?thx!
Can anyone explain to me this formula?
det(A)=∑ det(P)a1p(1)a2p(2)...a nP(n)
P
I understand the reasoning behind the formula, but i don't understand this notation...
Hello,
I would like to implement SAB=<psiApsiB>which is the overlap of two KohnSham determinants (psiA and psiB are two matrices containing each the molecular orbitals coefficients). Can anybody help me with this calculation? For case of SAA and SBB it is required to get the value 1...
Hello,
I would like to implement SAB=<psiApsiB>which is the overlap of two KohnSham determinants (psiA and psiB are two matrices containing each the molecular orbitals coefficients). Can anybody help me with this calculation? For case of SAA and SBB it is required to get the value 1 (perfect...
Hi, 2forms are defined as
du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}
But what if I have two concret 1forms in R^{3} like (2dx3dy+dz)\wedge (dx+2dydz) and then I calculte (2dx3dy+dz)\wedge...
Hi,
I have a rather trivial question but google did not really help me. So far I was always familiar with the fact that the determinant of a square matrix is positive.
But it is not. When I randomly execute det(randn(12)) in MATLAB I get a negative determinant every couple of trials...
What are the applications of a 4x4 determinant and what can it represent? To make it more clear, a 3x3 determinant represents vector perpendicular to two vectors and a 3x3 determinant can also be used to calculate torque. So, what can a 4x4 determinant do? Also, if there is a nxn determinant...
1st: Not a specific problem, I just didn't know where else to put it.
We just covered this today in class. Basically what we're doing is reducing higher level matrices to 2x2 matrices and using them to calculate the determinant.
I asked my teacher where that came from, and he was really...
I read that an equation of the form Ax=0 has a solution iff the matrix A has nontrivial Kernel, which makes sense as if A had trivial kernel then x would be trivial as well, meaning that only the x={0} solution would exist, right?
Secondly, I read that in order for A to have a nontrivial...
Homework Statement
If A is a 2x2 matrix, then det (2A * adj(A)^1) = ?
Homework Equations
Adj(A)A = det(A)I
The Attempt at a Solution
First, I separated them so it became det(2A) * det (1/ adj(A))
Then taking the 2 out, and it becomes 2^2, so 4 det(A) * det(1/ adj(A))
adj(A) =...