# 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

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Similarly, for a 3 × 3 matrix A, its determinant is

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{\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+cdh-ceg-bdi-afh.\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 n-dimensional volume of a n-dimensional 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.

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1. ### Mathematica Matrices in Mathematica -- How to calculate eigenvalues, eigenvectors, determinants and inverses?

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...
2. ### Exploring the Role of Determinants in Orienting Scientific Research

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.
3. ### POTW What is the Log Concavity of Determinants for Positive Definite Matrices?

Let ##A, B\in M_n(\mathbb{R})## be positive definite matrices. Prove that for ##0\le t\le 1##, $$\det(tA + (1 - t)B) \ge \det(A)^t\det(B)^{1-t}$$
4. ### I How can I convince myself that I can find the inverse of this matrix?

If I have a ##n\times n## matrix $$U= \begin{bmatrix} u_{11} & u_{12} &u_{13} & \cdots u_{1n} \\ 0 & u_{22} & u_{23} & \cdots u_{2n} \\ 0&0 &u_{33} &\cdots u_{3n}\\ \vdots & \vdots &\vdots & \cdots \vdots \\ 0 & 0 & 0 &\cdots u_{nn} \end{bmatrix}$$ Now, I don't want to use the fact that it's...
5. ### Proof of ##g(A_1, A_2, \cdots A_n) = c g (I_1, \cdots I_n)##.

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...
6. ### I How is uniqueness about the determinant proved by this theorem?

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...
7. ### I Why are determinants in 2x2 matrices and 3x3 matrices computed the way they are?

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...
8. ### A What physical meaning can the “determinant” of a divergency have?

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...

17. ### Proving that determinants aren't linear transformations?

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)##...
18. ### B Why the hate on determinants?

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 ?
19. ### Proving facts about matrices without determinants

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...
20. ### I Proving a result about invertibility without determinants

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...

23. ### Find inverse matrix using determinants and adjoints

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 =...
24. ### Show Tensor Determinants Equality

Homework Statement show that \det(\underline{\bf{A}})\det(\underline{\bf{B}}) = \det(\underline{\bf{AB}}) Homework Equations \begin{align*} &\underline{\bf{A}} = A_{ij} \underline{e}_i \otimes \underline{e}_j \\ &\underline{\bf{B}} = B_{mn} \underline{e}_m \otimes \underline{e}_n \\...
25. ### Matrix Determinants Homework: Finding the Answer

Homework Statement Homework EquationsThe Attempt at a Solution The answer in the solutions is given as : (2x+1)(x-1)(1-x), 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...
26. ### B Application of Matrices and Determinants

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...
27. ### What is the connection between cofactors, determinants, and matrix inverses?

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...
28. ### Slater determinants in Configuration Interaction

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...
29. ### Looking for insight into what the Determinant means....

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...

37. ### Factorizing determinants and rules to simplify them

Homework Statement 2. Evaluate the determinants ## \begin{vmatrix} 1 & 1 & 1\\ x & a & b \\ x^2 & a^2 & b^2 \\ \end{vmatrix} ## ## \begin{vmatrix} x & a & b \\ x^2 & a^2 & b^2 \\ x^3 & a^3 & b^3 \\ \end{vmatrix} ## and factorize both answers. Homework Equations Rules of determinants are...
38. ### Linear Algebra Determinants Proof

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...
39. ### Row/column operation on matrices and determinants

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?
40. ### Pfaffian and determinants of skew symmetric matrices

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...
41. ### Why is det(C)=det(A)^(n-1) for cofactors and determinants?

Any1 can explain to me why det(C)=det(A)^(n-1) where A is n-by-n matrix and C is the matrix of cofactors of A. I have been thinking, any 1 can help?thx!
42. ### Can you explain the determinant formula using permutation notation?

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...
43. ### Caculating SAB overlap of two Kohn-Sham determinants

Hello, I would like to implement SAB=<psiA|psiB>which is the overlap of two Kohn-Sham 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...
44. ### Caculating SAB overlap of two Kohn-Sham determinants

Hello, I would like to implement SAB=<psiA|psiB>which is the overlap of two Kohn-Sham 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...
45. ### What is the connection between 2-forms, determinants, and cross products in R^3?

Hi, 2-forms 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 1-forms in R^{3} like (2dx-3dy+dz)\wedge (dx+2dy-dz) and then I calculte (2dx-3dy+dz)\wedge...
46. ### Positive, negative, complex determinants

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...
47. ### Applications of 4x4 Determinant & Representations of nxn Determinants

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...
48. ### Reasoning behind determinants of high n square matrices

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...
49. ### Kernels and determinants of a matrix

I read that an equation of the form Ax=0 has a solution iff the matrix A has non-trivial 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 non-trivial...
50. ### What is the Determinant of a 2x2 Matrix Multiplied by its Adjoint Inverse?

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) =...