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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7
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2answers
198 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 ...
7
votes
2answers
123 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
1answer
116 views

Problem of determinant when $A^{-1}+B^{-1}=(A+B)^{-1}$

I have two $4\times 4$ real matrices $A$ and $B$, and it is known that $A^{-1}+B^{-1}=(A+B)^{-1}$ ($A$, $B$ and $A+B$ are invertible). How can I prove that $\det (A)=\det (B)$?
7
votes
1answer
202 views

Determinant of $2\times 2$ Block Matrix

I would like to know the proof for: The determinant of the block matrix\begin{pmatrix} A & B\\ C& D\end{pmatrix} equals $(D-1) \det(A) + \det(A-BC) = (D+1) \det(A) - \det(A+BC),$ when $A$ ...
7
votes
3answers
488 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
386 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
902 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
votes
2answers
121 views

Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.

Let $A$ be a $n\times n$ real matrix $A$ such that $A^2=-I$. Such an $A$ cannot be, Orthogonal. Invertible. Skew-symmetric. Symmetric. Diagonalizable. I tried to figure out the answer by looking ...
7
votes
2answers
119 views

A special case: determinant of a $n\times n$ matrix

I would like to solve for the determinant of a $n\times n$ matrix $V$ defined as: $$ V_{i,j}= \begin{cases} v_{i}+v_{j} & \text{if} & i \neq j \\[2mm] (2-\beta_{i}) v_{i} & \text{if} ...
7
votes
1answer
229 views

Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$

Let $M$ be a (real) $n \times n$ matrix. For $1 \leq i, j \leq n$ we denote by $M_{ij}$ the $(n-1) \times (n-1)$ matrix that we get when the $i$th row and $j$th column of $M$ are removed. Now, ...
7
votes
2answers
139 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
votes
2answers
297 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
2answers
150 views

Can this circulant determinant be zero?

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 ...
7
votes
2answers
310 views

Determinant of exact sequence

Let $0 \to A \to B \to C \to 0$ be an exact sequence of vector spaces. I want to show that I have a canonical isomorphism $$\det(B)= \det(A) \otimes \det(C).$$ Here, "det" refers to the $n$-th ...
7
votes
1answer
163 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
1answer
2k views

Proving determinant product rule combinatorially

One of definitions of the determinant is: $\det ({\mathbf C}) =\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}})$ I want to prove from this that ...
7
votes
1answer
680 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 ...
7
votes
3answers
284 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
216 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
44 views

Prove that p divides to algebraic multiplicity of the eigenvalue

I need help in the following exercise of a qualifying exam: Let $A$ be a matrix of size $m$ by $m$ over the finite field $\mathbb{F}_p$ such that $\operatorname{trace}\left(A^n\right)=0$ for all $n$. ...
7
votes
2answers
134 views

Problem involving trace and determinant of symmetric matrices

I've stumbled upon this exercise on a linear algebra book that asks me to determine all the ordered pairs $(a,b)$ of real numbers to which there exists an unique symmetric matrix $A\in R^{2\times 2}$ ...
7
votes
1answer
508 views

How to prove this inequality for determinant of Hermitian block matrix?

I am given an Hermitian positive definite matrix $$D=\left(\begin{matrix}A&\overline{C}^T\\C&B\end{matrix}\right)$$ $A$ and $B$ are square matrices. The task is to prove the following ...
7
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1answer
264 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
67 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
7
votes
1answer
273 views

Extending a Chebyshev-polynomial determinant identity

The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind: $$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
7
votes
1answer
172 views

Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
7
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1answer
124 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
3answers
91 views

If all entries of matrix $X$ are the same, then $\det (A+X)\det (A-X) \leq \det (A^2)$

I want to prove that $\det (A+X)\det (A-X) \leq \det (A^2)$ where $X $ is a matrix whose $n^2$ entries are all the same. I tried to write down the expressions involved but that didn't help me prove ...
7
votes
1answer
85 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
7
votes
1answer
122 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
0answers
241 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
7
votes
1answer
207 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
6
votes
5answers
350 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
9answers
361 views

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
6
votes
4answers
177 views

Find the determinant of $A + I$

Given a real valued matrix $A$ such that $A$ satisfies $AA^T = I$ and $\det(A)<0$, calculate $\det(A + I)$ My start : Since $A$ satisfies $AA^T = I$, $A$ is a unitary matrix. The determinant ...
6
votes
4answers
543 views

Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
6
votes
2answers
2k 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 ...
6
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5answers
475 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
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4answers
711 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
6
votes
4answers
471 views

What is the intuitive meaning of a determinant? [duplicate]

I know how to calculate a determinant, but I wanted to know what the meaning of a determinant is? So how could I explain to a child, what a determinant actually is. Could I think of it as a measure ...
6
votes
3answers
19k views

Determinant of symmetric matrix

Given the following matrix, is there a way to compute the determinant other than using laplace till there're $3\times3$ determinants? \begin{pmatrix} 2 & 1 &1 &1&1 \\ 1 & 2 ...
6
votes
4answers
385 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
6
votes
2answers
200 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
109 views

If $A^n = I$, $n$ odd, $A$ a square integer matrix, does $A = I$?

Edit: Crap, even my hypothesis was wrong. If you put $A = \left[ \begin{array}{cc} 1&-1\\3&-2 \end{array} \right]$, then $A^3 = I$ but no eigenvalue is $1$. (What's true is that all ...
6
votes
2answers
427 views

How can I quickly find the determinant of this matrix

$$ \begin{vmatrix} 14 & 2 & 1 & 3\\ 31 & 4 & 5 & 6\\ 26 & 3 & 7 & 4\\ 10 & 1 & 3 & 2\\ \end{vmatrix} ...
6
votes
3answers
503 views

Question about determinants

I am working on some practice problems and I'm unsure where to begin this problem. It starts off by giving $\det(X)= 1$ for the following matrix $X$:$$ \begin{matrix} a & 1 & d \\ b & 1 ...
6
votes
2answers
542 views

Determinant always equal to zero?

I just finished writing a computer program that takes as input a number of matrices and computes the inverse of the product of matrices. To test this program, I wanted to input a 3x2 matrix followed ...
6
votes
5answers
447 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 ...
6
votes
3answers
228 views

Formal proof of $\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i)$

I'm looking for a formal proof for: $$\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i).$$ I'm very new to matrix theory therefore please forgive me if you find this elementary. Your help in this ...
6
votes
1answer
837 views

Why is the determinant invariant under row and column operations?

I know that we may add any row to any other in a determinant and its value remains the same. This is clear enough since elementary matrices corresponding to row and column operations have determinant ...