# 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

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{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}
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. ### Linear Algebra Determinant proof

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...
2. ### POTW Inequality of Determinants

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

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
4. ### Decompose 4x4 determinant into 24 determinants -- How many are zero?

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...
5. ### Fortran Does Fortran have a built-in function to calculate the determinant?

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
6. ### I Determinant of a specific, symmetric Toeplitz matrix

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...
7. ### 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...
8. ### I Contracted Christoffel symbols in terms determinant(?) of metric

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...
9. ### B Determinant of a transposed matrix

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...

12. ### Finding roots and complex roots of a determinant

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...
13. ### I Showing Determinant of Metric Tensor is a Tensor Density

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...
14. ### I When is it better to call a thing "modulus" and when "determinant"?

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"?
15. ### 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...
16. ### I How to obtain the determinant of the Curl in cylindrical coordinates?

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}...
17. ### Det of Triangular-like Matrix & getting stuck in Algorithmic Proof

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...
18. ### How to find the determinant of this matrix?

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...
19. ### MHB 311.3.2.16 Find the determinant with variables a b c d e f g h i

$\tiny{311.3.2.16}$ 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...
20. ### Finding the determinant of a matrix using determinant properties

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...
21. ### Using a determinant to find the area of the triangle (deriving the formula)

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...
22. ### Finding the Determinant to find out if the matrix is invertible

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...
23. ### A Functional Determinant of a system of differential operators?

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...
24. ### Product of two Kronecker delta symbols as a determinant

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.
25. ### MHB Proving the Formula for Determinants by Induction

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...
26. ### MHB Test for compatibility of equations - Determinant |A b|

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)
27. ### MHB How Does the Determinant of a Matrix Relate to the Area of a Parallelogram?

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...
28. ### A Stuck on evaluating this functional determinant

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...
29. ### MHB [ASK] Determinant of a Matrix with Polynomial Elements

Help me if what I have done so far can be simplified further.
30. ### Why is my calculation for the determinant of a matrix incorrect?

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.
31. ### A I can't verify a relationship between cofactor and determinant

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.
32. ### I Time evolution of a Jacobian determinant

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...
33. ### I Calculating Jacobian Determinant

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...
34. ### MHB 290 Expanding this determinant about the the second column....

where does 32 (red) come from ? nevermind looks its (-8)(4)=-32 but will probable have more ? on this example
35. ### Advice on calculating the determinant for 3x3 Matrix by inspection

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...
36. ### MHB E1.4b Determinant with zero column

$$\left[\begin{array}{rrrrr} 1 &0 &2 &1\\ 1 &1 &0 &1\\ 1 &3 &4 &1\\ -1 &-3 &-4 &-1 \end{array}\right]=\color{red}{0}$$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...