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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277 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
1answer
138 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 ...
8
votes
2answers
396 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
votes
2answers
228 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
179 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
votes
3answers
304 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
104 views

How to show a Determinantal inequality

If $A, B$ and $C$ are $n\times n$ positive semidefinite matrices. How to show that $$\det(A + B) + \det(A + C)\le \det A + \det(A + B + C)?$$
7
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5answers
905 views

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 ...
7
votes
5answers
294 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 ...
7
votes
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
6answers
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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 ...
7
votes
5answers
176 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
votes
2answers
1k 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 ...
7
votes
2answers
102 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
votes
1answer
148 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
votes
1answer
347 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
votes
2answers
114 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
210 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
votes
2answers
150 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
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1answer
100 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
votes
3answers
213 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
146 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
votes
1answer
221 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
113 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
votes
1answer
110 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
116 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
3answers
228 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
7
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0answers
157 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
6
votes
5answers
323 views

How to prove $I + t X$ is invertiable for small enough $ | t | ?$

Let $X \in \text{GL}_n(\mathbb{R})$ be an arbitrary real $n\times n$ matrix. How can we prove rigorously: $$ \underset{b>0} {\exists} : \underset{|t|\le b} {\forall} : \det (I + t X) \neq 0 $$ If ...
6
votes
5answers
198 views

If $A^T=-A$, then A is not invertible

Let $n \in \mathbb{N}$ be odd and $A \in$Mat$(n,\mathbb{R})$ with $A^T=-A$. Show that $A$ is not invertible. I have no idea how to start this...
6
votes
2answers
251 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
6
votes
2answers
183 views

Why this determinant is conformally invariant?

While I was reading a paper about random analytic function I found a statement that I was not able to prove and after try brute force and search for some references I decided to ask for a help here. ...
6
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1answer
11k 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 ...
6
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2answers
855 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 ...
6
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2answers
2k views

The determinant of block triangular matrix as product of determinants of diagonal blocks

I am given the following partitioned - upper-triangular matrix: $$ \begin{bmatrix} A_1 &* &* &* &* &* \\ 0& A_2 &* &* &* &* \\ .& 0& ...
6
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2answers
135 views

Explanation of a cross product result

In my book the result $$(u\times v)\cdot(x\times y)=\begin{vmatrix} u\cdot x & v\cdot x \\u \cdot y & v \cdot y\end{vmatrix},$$ where u, v, x and y are arbitrary vectors, is stated (here ...
6
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4answers
537 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 ...
6
votes
1answer
180 views

How to deduce that there are $(n^3+2n-3)/3$ multiplications for the determinant evaluation?

In Friedberg's Linear Algebra, the author points out that the evaluation of the determinant of an $n\times n$ matrix by cofactor expansion along any row requires over $n!$ multiplications, whereas ...
6
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2answers
130 views

Just a trivial doubt on linear algebra

The question is: If $a,b,c$ are negative distinct real numbers,then the determinant $$ \begin{vmatrix} a & b & c \\ b & c & a\\ c & a & b \end{vmatrix} $$ is $$(a) \le 0 ...
6
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2answers
1k views

Vandermonde determinant by induction

The determinant at the top-left of the page can be done by induction, it says show that. I have done this before, if I submit this will I get marks? MORE IMPORTANTLY how do I do it by induction? The ...
6
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2answers
133 views

What is the maximum possible value of determinant of a matrix whose entries either 0 or 1?

My question is simply the title: What is the maximum possible value of determinant of a matrix whose entries either 0 or 1 ?
6
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1answer
419 views

Solving $n$-queens with determinants

I keep reading about a proposed method of finding solutions to the $n$-queens problem using determinants, but I can't find any specific details anywhere. Can somebody explain to me how to find ...
6
votes
1answer
146 views

Determinant of $A + \epsilon X$

In the Wikipedia article on the determinant, it is stated that $$\det \left ( A + \epsilon X \right ) - \det \left ( A \right ) = {\rm tr} \left ( {\rm adj} \left ( A \right ) X \right ) \epsilon + ...
6
votes
1answer
217 views

Do multiplicative maps of matrices factor through determinants?

Given a map $f:M_n(k)\to k$ (with $k$ some field) such that $f(AB)=f(A)f(B)$ for all matrices $A$ and $B$, is it necessarily the case that $f$ factors through the determinant, i.e. does there exist a ...
6
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1answer
252 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 ...
6
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0answers
73 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
6
votes
1answer
130 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
5
votes
5answers
504 views

Is a square matrix whose diagonal and antidiagonal elements are all zero always singular?

Consider an $n\times n$ matrix whose primary and secondary diagonal elements are all zero. Does it necessarily follow that the determinant vanishes for these matrices? When $n=1,2,3,4$, the matrix is ...
5
votes
2answers
511 views

Why must the determinant of a matrix with with integer entries be an integer?

Why must the determinant of a matrix with integer entries be an integer? Note: I know what a determinant of a matrix is, not sure how to explain this question. Is that because if the matrix is made ...
5
votes
4answers
970 views

$\det(A + B) = \det(A) + \det(B)$?

Well considering two $n \times n$ matrices does the following hold true: $$\det(A+B) = \det(A) + \det(B)$$ Can there be said anything about $\det(A+B)$? If $A/B$ are symmetric (or maybe even of the ...