Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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260
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10answers
34k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
62
votes
3answers
2k views

Cute Determinant Question

I stumbled across the following problem and found it cute. Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant: ...
31
votes
3answers
2k views

How to find the determinant of this matrix?

Today at my linear algebra exam, there was this question that I couldn't solve. There was a matrix $A$ $$A=\begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & ...
31
votes
2answers
2k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
30
votes
2answers
8k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
30
votes
2answers
783 views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = ...
26
votes
1answer
693 views

How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$

Question: let $a_{i}>1,i=1,2,3,\cdots,n$,and such $a_{i}\neq a_{j}$,for any $i\neq j$ define the matrix $$A=\left(\dfrac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}$$ show that: ...
25
votes
1answer
837 views

Prove the determinant of this matrix

We have a square matrix, that all elements on main diagonal are zero, and other elements are following: $$a_{i,j}=\begin{cases} 1,&\text{if i+j belongs to Fibonacci numbers,}\\ 0,&\text{if ...
24
votes
9answers
18k views

How to show that $\det(AB) =\det(A)\det(B)$

Given two square matrices $A$ and $B$, how do you show that $\det(AB) = \det(A)\det(B)$, where $\det(\cdot)$ is the determinant of the matrix?
21
votes
3answers
579 views

If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$

Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$ Prove that $$\det(A+B)=\det B.$$
21
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4answers
1k views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
21
votes
2answers
418 views

Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer

Let $x_1,...,x_n$ be distinct integers. Prove that $$\prod_{i<j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$ I know there is a solution using determinant of a matrix, but I can't remember it now. Any ...
19
votes
2answers
2k views

Is it always true that $\det(A^2+B^2)\geq0$?

Let $A$ and $B$ be real square matrices of the same size. Is it true that $$\det(A^2+B^2)\geq0\,?$$ If $AB=BA$ then the answer is positive: ...
19
votes
10answers
7k views

Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
19
votes
3answers
1k views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
19
votes
2answers
349 views

How find this determinant $\det(\cos^4{(i-j)})_{n\times n}$

Question: Define the matrix $A_{k}=(a^k_{ij})_{n\times n}\quad$where $a_{ij}=\cos{(i-j)},\quad n\ge 6$ Find the value $$\det(A_{4})=\:?$$ My try: since $$\det(A_{4})=\begin{vmatrix} ...
18
votes
4answers
978 views

A problem on Condition $\det(A+B)=\det(A)+\det(B)$

Let $A$ be a matrix $n\times n$ matrix that for any matrix $B$ we have $\det(A+B)=\det(A)+\det(B)$. If this imply that $A=0$? or $\det(A)=0$?
17
votes
2answers
513 views

Why is there no generalization of the determinant to infinite dimensional vector spaces?

This question is to add to my understanding why the concept of a determinant does not extend to an infinite dimensional vector space. I am already aware of a couple facts which hint why this is so: ...
16
votes
2answers
1k views

Slick proof the determinant is an irreducible polynomial

A polynomial $p$ over a field $k$ is called irreducible if $p=fg$ for polynomials $f,g$ implies $f$ or $g$ are constant. One can consider the determinant of an $n\times n$ matrix to be a polynomial in ...
16
votes
4answers
701 views

Determinant of a generalized Pascal matrix

Let $M$ denote the infinite matrix defined recursively by $$ M_{ij} = \begin{cases} 1, & \text{if } i=1 \text{ and } j=1; \\ aM_{i-1,j}+bM_{i,j-1}+cM_{i-1,j-1}, & ...
15
votes
9answers
996 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
15
votes
4answers
213 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
15
votes
2answers
594 views

easier way of calculating the determinant for this matrix

I have to calculate the determinant of this matrix: $$ \begin{pmatrix} a&b&c&d\\b&c&d&a\\c&d&a&b\\d&a&b&c \end{pmatrix} $$ Is there an easier way of ...
15
votes
2answers
246 views

Help deriving that $\mathrm{sign} : S_n\to \{\pm 1\}$ is multiplicative

$\def\sign{\operatorname{sign}}$ For homework, I am trying to show that $\sign:S_n \to \{\pm 1\}$ is multiplicative, i.e. that for any permutations $\sigma_1,\sigma_2$ we have $$\sign(\sigma_1 ...
14
votes
6answers
66k views

What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
14
votes
2answers
485 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
14
votes
1answer
4k views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
14
votes
2answers
179 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
13
votes
4answers
726 views

Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$. My conjecture is ...
13
votes
3answers
609 views

Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$

I have to prove/disprove this: If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$ I believe it is true but I can not think of a direct way to prove it. Any ...
13
votes
3answers
235 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
13
votes
2answers
2k views

What is the origin of the determinant in linear algebra?

We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible ...
13
votes
2answers
223 views

Evaluate the determinant $\det\left[ \binom{2n}{n+i-j} \right]_{i,j=0}^{n-1}$

I am trying to show that: \begin{equation} \det\left[ \binom{2n}{n+i-j} \right]_{i,j=0}^{n-1}=\prod_{i=0}^{n-1} \frac{\binom{2n+i}{n}}{\binom{n+i}{n}} \end{equation} I have tried playing with the ...
13
votes
4answers
240 views

Find the determinant of $A$ satisfying $A^{-1}=I-2A.$

I am stuck with the following problem: Let $A$ be a $3\times 3$ matrix over real numbers satisfying $A^{-1}=I-2A.$ Then find the value of det$(A).$ I do not know how to proceed. Can someone point me ...
13
votes
1answer
216 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
13
votes
3answers
459 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
13
votes
1answer
463 views

How to prove a Wronskian identity?

The following Wronskian identity can be proved by expanding both sides and checking that two sides are the same. But how to prove it more elegantly? Let $u_1(x), u_2(x), u_3(x), u_4(x)$ be four ...
13
votes
1answer
330 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
13
votes
0answers
469 views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for ...
12
votes
4answers
930 views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ ...
12
votes
5answers
595 views

Show determinant of matrix is non-zero

I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help ...
12
votes
3answers
242 views

determinant inequality $\det(A^2+AB+B^2)\geq\det(AB-BA)$

$A,B$ are two $2\times 2$ real matrices, then $$\det(A^2+AB+B^2)\geq\det(AB-BA)$$ The inequality is equivalent to the following problem: Let $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
12
votes
2answers
1k views

The Hessian of the Determinant

It is well known how to take the derivative of the determinant: let $A(s)$ be a family of square matrices smoothly parametrised by the variable $s$ (in other words, $A:\mathbb{R}\to \mathbb{R}^{N^2}$ ...
12
votes
1answer
256 views

Limit of a sequence of determinants.

Let $\beta>0$ be given. For each $n\geq 2$, let $\Delta_n=\det M_n$ denote the determinant of the following matrix: \begin{align} M_n = \begin{pmatrix} 2+\epsilon^2 & -1 & 0 & 0 ...
12
votes
1answer
370 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
11
votes
4answers
3k views

What is the fastest way to find the characteristic polynomial of a matrix?

Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$. I know that: the coefficient of $\lambda^n$ is $(-1)^n$, the coefficient of ...
11
votes
3answers
390 views

Determinant of Abstract Matrix

Given an $n \times n$ matrix $A$, where $x$ is any real number: $A = \left[ \begin{array}{ c c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & x ...
11
votes
3answers
4k views

How do I prove that $\det A= \det A^t$?

I found this exercise in Artin. It asks me to prove that $\det A= \det A^t$ where $A^t$ is the transpose of the matrix $A$. Can anyone please comment whether my proof is correct or not? Attempted ...
11
votes
2answers
694 views

Do determinants of binary matrices form a set of consecutive numbers?

While pondering a solution for the problem of generating random 0-1 matrices with small absolute determinants, I once again realise how little I know about 0-1 matrices. My initial idea was to pick a ...
11
votes
1answer
131 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...