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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2answers
538 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 ...
10
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2answers
1k 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 ...
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2answers
138 views

Prove or disprove : $\det(A^k + B^k) \geq 0$

This question came from here. As the OP hasn't edited his question and I really want the answer, I'm adding my thoughts. Let $A, B$ be two real $n\times n$ matrices that commute and $\det(A + ...
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0answers
373 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 ...
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votes
6answers
32k 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 ...
9
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6answers
701 views

Determinant of a special skew-symmetric matrix

Simple calculation show that: $$ \begin{align} \det(A_2)=\begin{vmatrix} 0& 1 \\ -1& 0 \end{vmatrix}&=1\\ \det(A_4)=\begin{vmatrix} 0& 1 &1 &1 \\ ...
9
votes
4answers
1k 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 ...
9
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2answers
266 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
9
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2answers
348 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
8
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4answers
890 views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
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7answers
628 views

Determinant of a specially structured matrix

I have the following $n\times n$ matrix: $$A=\begin{bmatrix}a&b&\cdots&b\\b&a&\cdots&b\\\vdots& &\ddots&\vdots\\b&\cdots&b&a\end{bmatrix}$$ where $0 ...
8
votes
3answers
564 views

The determinant function is the only one satisfying the conditions

How can I prove that the determinant function satisfying the following properties is unique: $\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and ...
8
votes
3answers
87 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
8
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4answers
310 views

What does it mean if $\det(A)$ equals $1$?

What does it mean if $\det(A)$ equals $1$? Does it mean that the identity matrix can be obtained from $A$ by only adding multiples of rows onto others?
8
votes
2answers
252 views

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
8
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1answer
139 views

Is there a name referring to this result?

For any real $m \times n$ matrix $A$, it seems that $$\det(I_n + A^{T}A) = \det(I_m + AA^{T}) $$ always holds, where $I_n$ is the identity matrix of size $n$. Though I have not tried to prove this ...
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2answers
452 views

Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd ...
8
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2answers
236 views

Probability of determinants being coprime

I have a question that is not of particular significance, but I would love to understand the underlying principles. Suppose we have two square 3x3 matrices, $M_1$ and $M_2$ with $$M_1 = ...
8
votes
2answers
182 views

Positivity of a determinant

I'm stuck to prove the following exercise : Given real numbers $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$, show that $$ \det(e^{\large{x_iy_j}})_{i,j=1}^n>0 $$ provided that $x_1<\cdots<x_n$ and ...
8
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3answers
317 views

Without choosing bases, how to show that the determinant is multiplicative in this sense?

I was recently considering this statement: Let $V$ be a finite-dimensional $k$-vector space, and let $\phi:V\to V$ be an endomorphism. Suppose that $W\subseteq V$ is a subspace that is stable ...
8
votes
2answers
108 views

How to show a Determinantal inequality

If $A, B$ and $C$ are $n\times n$ positive semidefinite matrices. How to show that $$\det(A + B) + \det(A + C)\le \det A + \det(A + B + C)?$$
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5answers
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Showing determinants using trace in a 2x2 matrix

I am confused about this homework question. It says "Show that : $\det(A) = \frac 12 \begin{vmatrix}\operatorname{tr}(A)&1\\\operatorname{tr}(A^2)& \operatorname{tr}(A)\end{vmatrix}$ for ...
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5answers
372 views

Determine the value of a second determinant based on the first

I know the theory of determinants, but I have no idea how to apply it to this problem. Suppose $$\det\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} = 6$$ What is the value ...
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4answers
2k views

How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
7
votes
4answers
260 views

Determinant of a Special Symmetric Matrix

If $A$ is a symmatric matrix of odd order with integer entries and the diagonal entries $0$ then $A$ has determinant value even. I can prove the result if I can show that the eigenvalues of $A$ are ...
7
votes
3answers
281 views

Is $\;\det(A^n) =\left(\det (A)\right)^n\;$?

How can the value of $\;\det\left(A^{11}\right)\;$ be calculated from $\;\det(A)$? Generally how can $\;\det\left(A^n\right)\;$ be obtained from $\;\det(A)$?
7
votes
2answers
273 views

Circulant determinants

Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true? There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the ...
7
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2answers
1k views

Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant

If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$? I remember seeing such a ...
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1answer
13k views

Using the Determinant to verify Linear Independence, Span and Basis

Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a ...
7
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2answers
979 views

where did determinant come from? [duplicate]

Possible Duplicate: What's an intuitive way to think about the determinant? I just learned the basics of matrices. Then I came across the magical formula $$\det(AB)=\det(A)\det(B)$$ I ...
7
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5answers
208 views

Calculate the determinant of the $2n \times 2n$ matrix with entries equal to zero on the main diagonal, $1$ below and $-1$ above [duplicate]

Calculate the determinant of the $2n \times 2n$ matrix with entries equal to zero on the main diagonal, equal to $1$ below and equal to $-1$ above. I'll denote this matrix $A_{2n}$. So for example ...
7
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1answer
152 views

How to show that $\mathrm{SL}(2,\mathbb Z) = \langle A, B\rangle$?

Show, that if $\mathbf{A}= \left( \begin{array}{cc} 1&1\\ 0&1 \end{array} \right)$, $\mathbf{B}= \left( \begin{array}{cc} 0&1\\ -1&0 \end{array} \right)$ and $\mathrm{SL}(2, ...
7
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2answers
2k views

Use of determinants

I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a ...
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2answers
111 views

How to evaluate this determinant?

Can someone give me a hint how to solve $$\left|\begin{array}{ccccc} 1 & 1 & \ldots & & 1\\ 2x_{1} & 2x_{2} & & & 2x_{n}\\ \vdots\\ nx_{1}^{n-1} & ...
7
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1answer
357 views

Is this a well known determinant identity? Are there any generalizations?

Let $A$ be a $3\times3$ matrix and for any $i,j\subseteq\{1,2,3\}$, let $A^{i,j}$ denote the $2\times2$ matrix resulting from removing row $i$ and column $j$ from $A$. Then: ...
7
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2answers
116 views

Determinant of matrix $(x_j^{n-i}- x_j^{2n-i})_{i,j=1}^{n}$

Good evening all, I am determined to determine this determinant: $$D = \det{\left[x_j^{n-i} - x_j^{2n-i}\right]_{i,j=1}^{n}}$$ Looking at the smaller cases, leads me to believe that $$D = \prod_{1 ...
7
votes
3answers
247 views

Matrix Determinant Identity

I have come across an observation about the determinant of a matrix, but I don't know how to prove it in general. Let me demonstrate it through an example. $$ \begin{align} \left| \begin{matrix} 1 ...
7
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2answers
261 views

Determinant of a linear map given by conjugation

Suppose we have two fields $K\subset L$ and $A\in GL(n,L)$. See $L^{n\times n}$ as a $K$-vectorspace, then $$C_A\colon L^{n\times n}\rightarrow L^{n\times n},B\mapsto ABA^{-1}$$ is a $K$-linear map. ...
7
votes
4answers
587 views

Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$

Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
7
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2answers
103 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
7
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172 views

Historical meaning and usage of determinant

Can anyone please explain how, why, and where determinants were developed/formalized? What was their historical usage? Why were they initially formulated and what were they used for (and later ...
7
votes
1answer
126 views

Prove that determinant of matrix equal $\pm1$ or $0$

We are given square binary matrix $A_n$. Data contained by A comply the following rule: if row has any 1's then they would appear there only successively (row $(1\space 1\space0\space1 )$ is ...
7
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2answers
109 views

trace , determinant and which of the following are true(NBHM-$2014$)

Let $A \in M_2(\mathbb R)$ be a matrix which is not a diagonal matrix . Which of the following statements are true?? a. If $tr(A)=-1$ and $detA=1$, then $A^3=I$. b. If $A^3=I$, then ...
7
votes
3answers
228 views

Singular covariance matrix

I am looking into the process $\{X_t, t\in\mathbb{Z}\}$, $X_t=A\cos(\lambda t)+B\sin(\lambda t)$, here $\lambda\in(0,\pi)$ is fixed, $A$ and $B$ are uncorrelated random variables with $EA=EB=0$, ...
7
votes
1answer
164 views

Determinant of the transpose via exterior products

Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant: If $t:U \to U$ is a linear operator and $\dim(U)=n$ then ...
7
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1answer
246 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
7
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1answer
129 views

How to prove this $A$ is an invertible matrix

let Symmetric matrix $A=(a_{ij})_{n\times n},n\ge 2$,and $$\begin{cases} a_{jk}=j+k\cdot i&j< k\\ a_{jj}=2j\cdot(i+1) \end{cases}$$ where $i^2=-1$ show that :$A$ is Invertible matrix My ...
7
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1answer
114 views

Determinant of matrices along a line between two given matrices

The question, with no simplifications or motivation: Let $A$ and $B$ be square matrices of the same size (with real or complex coefficients). What is the most reasonable formula one can find for ...
7
votes
1answer
117 views

What is the limit $\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le m/n\le r/s}x^my^n$?

Let $S=[0,1)^2$ and $m,n$ are positive integers and $p/q,r/s$ are positive rationals with $p/q<r/s$. What is the limit $$\lim\limits_{(x,y)\to(1,1),\ (x,y)\in S}(1-x^py^q)(1-x^ry^s)\sum_{p/q\le ...
7
votes
2answers
111 views

Matrix with determinant 0

If $A \in M_3(\mathbb{R})$ is a $3 \times 3$ matrix with $\det(A)=0$ and the square of each element equals its cofactor, do we necessarily have $A=0_3$? $a_{ij}^2=A_{ij}$, where ...