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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98 views

How to show that $\det(S+dd^T)=\det S\cdot(1+d^TS^{-1}d)$?

Let $x_i,\overline x, d$ (with $i\leq n$) be given vectors (column, of the same length). Put $S=\frac{1}{n}\sum_{i=1}^n(x_i-\overline x)(x_i-\overline x)^T$. Assume that $S$ is invertible. Is the ...
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1answer
116 views

Determinant of an $n\times n$ matrix with 5's on the diagonal and 2's on the superdiagonal and subdiagonal [duplicate]

Possible Duplicate: Special determinant formula for a specific matrix How to find $\det A_n$ as a function of $n$? $$A_n=\begin{pmatrix} 5&2 &0& 0 & \ldots & 0\\ ...
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3answers
436 views

How to efficiently compute the determinant of a matrix using elementary operations?

Need help to compute $\det A$ where $$A=\left(\begin{matrix}36&60&72&37\\43&71&78&34\\44&69&73&32\\30&50&65&38\end{matrix} \right)$$ How would one use ...
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2answers
313 views

Determinant of a block matrix with $\mathrm{Id}$ and $0$ in the diagonal

How to compute the determinant $\det A$ depending on $B$ and $C$, where $$ A = \left(\begin{matrix}\mathrm{Id} & B \\ C & 0 \end{matrix} \right), $$ a) when $C$ is square, b) $C$ has more ...
2
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1answer
216 views

How to show that the determinant of $A$ is non-zero?

Let $A$ be an $n$ by $n$ real matrix such that all entries not on the diagonal are positive, and the sum of the entries in each row is negative. How to show that the determinant of $A$ is non-zero?
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0answers
462 views

Determinant of a symmetric matrix values in each column and row don't repeat

Could you help me count the determinant of this symmetric matrix? $\begin{vmatrix} a&2b&3c&6d\\b&a&3d&3c\\c&2d&a&2b\\d&c&b&a\end{vmatrix} $
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1answer
161 views

Use row reduction to prove that $\det(\mathbf{A})=\det(\mathbf{A}^{T})$

I need to prove that the determinant of a matrix is equal to the determinant of its transpose. This fact is obviously easy to prove using the definition of the determinant, but the question stipulates ...
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1answer
66 views

Errors while calculating the unknown of a matrix?

I am currently facing a problem for calculating the unknown in a matrix: The Determinant is $A=35$ and the matrix is $$A= \begin{bmatrix} 7 & 8 & 6 & u \\ -5 & 8 & 6 ...
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1answer
45 views

Matrix determinant $b_{ij}=c^{i-j}a_{ij}$

I found this problem in A. Kostrikin's algebra book. There is no solution or a hint to it there. Only answer: $\det B=a$. Let $A = [a_{ij}] \in \mathcal{M}(n,n; K), \ \det A=a, \ \ c \in K, c \neq0$ ...
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0answers
47 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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4answers
85 views

Determinant of a $4\times4$ matrice with one unknown?

I have to calculate the determinant of this matrice. I want to use the rule of sarrus, but this does only work with a $3\times3$ matrice: $$ A= \begin{bmatrix} 1 & -2 & -6 & u \\ ...
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2answers
358 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 ...
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1answer
58 views

Proof of the naturality of integration

I have a bit of a problem with the following identity: Suppose that $U, V \subset \mathbb{R}^n$, are two open sets. Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. ...
7
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3answers
232 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
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2answers
266 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. ...
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576 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 ...
3
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3answers
703 views

How to randomly construct a square full-ranked matrix with low determinant?

How to randomly construct a square (1000*1000) full-ranked matrix with low determinant? I have tried the following method, but it failed. In MATLAB, I just use: n=100; A=randi([0 1], n, n); while ...
2
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1answer
280 views

How to show a certain determinant is non-zero

For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant where $\lambda_1 \lt \lambda_2 \lt \ldots ...
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4answers
472 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
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1answer
176 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 + ...
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1answer
136 views

expressing product as Vandermonde determinants

Is it possible to express the product: $$ \frac{\prod_{i < j} (a_i - a_j)(b_i - b_j) }{\prod_{i,j} (a_i - b_j) }$$ as the determinant of a single matrix ? This comes from a physics paper. Should ...
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4answers
282 views

Special determinant formula for a specific matrix

How to show that the determinant of the following $(n\times n)$ matrix $$ \begin{pmatrix} 5 & 2 & 0 &0&0&\cdots & 0\\ 2 & 5 & 2 & ...
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2answers
3k 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& ...
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1answer
231 views

Zeroes in a 3x3 Matrix Determinant

My professor found the cubic roots of a 3x3 matrix by doing the following. I don't understand how step 2 came about and why he applied the same for step 4 on row 1 instead of row 2. Step 1: ...
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1answer
193 views

Help with understanding the general formula for the determinant? [duplicate]

Possible Duplicate: What’s an intuitive way to think about the determinant? Could anyone give an intuitive explanation of the determinant? I know mostly what the determinant means and I can ...
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4answers
225 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 ...
7
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178 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 ...
2
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0answers
89 views

Matrix inversion is to determinants as matrix logarithm is to what?

I have not put much effort into this question but I have thought about it for a year or so. Is there such thing as a "logarithmic determinant"? The starting point for this is that the determinant of ...
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2answers
265 views

Eigenvalues, minimal polynomials and characteristic equation

What is the difference between solving $\det(xI- A) = 0$ and $\det(A-\lambda I) = 0$ to find eigenvalues of a Matrix $A$? Is the only difference that the first equation will give you the ...
0
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2answers
156 views

Find the determinant

I am trying to find the eigenvalues of a matrix and I cannot remember how to find the determinant of $A-\lambda I$: \begin{equation} \pmatrix{1-\lambda& 2& 1 \\ 2& -\lambda & ...
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4answers
233 views

Matrix Determinant

So I'm reading through my linear algebra textbook to review for my final, and happened upon this statement: The determinant of a matrix with positive entries must be positive. Off the top of my ...
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3answers
157 views

Prove an identity including determinant

Prove that: $$\begin{equation} \begin{vmatrix} x_0^{2n+1}&x_0^{2n}&\cdots&x_0&1\\ x_1^{2n+1}&x_1^{2n}&\cdots&x_1&1\\ ...
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3answers
599 views

Determinants of block matrices

Let $A,B \in \mathbb{R}^{n,n}$. Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$. Show that $\det(C) \in \mathbb{R}$ ...
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2answers
905 views

derivative of a determinant of a matrix with respect to an element that appears many times in the matrix

I've been trying to find material on matrix calculus but it seems hard to find ones with understandable proofs. I'm doing research work and I am trying to verify some computation. Suppose that I have ...
2
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1answer
81 views

The derivative of characterestic polynomial?

Let $A\in M_{n}(R)$ and $f(x)$ be the characterestic polynomial of $A$. Is it true that $f'(x)=\sum_{i=1}^{^{n}}\sum_{j=1}^{n}\det(xI-A(i\mid j))$ which $A(i\mid j)$ is a submatrix of $A$ obtained by ...
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2answers
607 views

Row Reduction with Cofactor Expansion

My calculator says the determinant of $$\begin{pmatrix}3 &0&6&-3\\0&2&3&0\\-4&-7&2&0\\2&0&1&10\end{pmatrix}$$ is $396$. However, the website I got the ...
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3answers
340 views

Minimal polynomial, determinants and invertibility

I need to prove: if a matrix $A$ is invertible, then the minimal polynomial $m_a(0) \neq 0$ There is one definition I am unsure of or need help making more clear. I will proceed with proof by ...
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5answers
3k views

For $det(A)=0$, how do we know if A has no solution or infinitely many solutions?

If the determinant det(A) of the matrix A of a non-homogeneous system of equations is 0, then how do we know if it has no solutions or infinitely many solutions? And while we are at it, kindly answer ...
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1answer
117 views

Prove equivalence between the determinant of a matrix and the product of specific submatrices

Proposition Let $A\in\mathfrak{M}_{(m+n)\times(m+n)}(\mathbb{K})$, $B\in\mathfrak{M}_{n}(\mathbb{K})$, $D\in\mathfrak{M}_{m}(\mathbb{K})$. If $$A=\left(\begin{array}{cc}B& C\\0 & D ...
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1answer
87 views

Determinant of $M^T M$ for sparse matrixes

I need to compute numerically the determinant of $M^T M$ where $M$ is a large (non-square) sparse matrix. Is there any sensible way to compute it? More generally, is there any mathematical property ...
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1answer
92 views

Linear algebra preminary theorem

I'm self studying linear algebra from the book by L.Mersky and struggling with the below theorem. If $(λ1,....λn),(μ1,...μn)$ and $(K1,....Kn)$ are arrangements of $(1,....n)$ then 1. The proof ...
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2answers
206 views

How to show this determinant $D \not= 0$ (EDIT) maybe figure out is impossible

SORRY, I made a typo. it should be $D \not= 0$,not $D>0$. It is a bit like Vandermonde determinant $$D=$$ $$\begin{vmatrix} 1 & 2 & 3&\cdots &2008&2009 & 2010 & 2011\\ ...
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1answer
101 views

Invertibility of matrix with each element equal to cofactor

I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
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2answers
126 views

Trying to find $\det (B)$

Let $A=\left(\begin{matrix}1&1&-1\\-1&1&1\\1&-1&1\end{matrix} \right)$,and ${A}^{T}B{\left( \cfrac{1}{2}{A}^{T}\right)}^{T}-8{A}^{-1}B=I$, How to compute $\left|B \right|$?。
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3answers
92 views

Determinant of a $4\times4$ invertible matrix

Let $A$ be a $4$ by $4$ invertible matrix, such that $\det(3A)=3\det(A^4)$. Then $\det(A)=3$. Would somebody please give me some clues on this? Thanks
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1answer
811 views

Unitary matrix proof

Prove that unitary matrix $U$ satisfies $|\det U| = 1$, but $\det U$ is different from $\det U^{H}$. How can I prove these two statements? I guess I should use the fact that every column of unitary ...
3
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4answers
179 views

Value of Vandermonde type determinant

Let $x_1,...,x_n $ are distinct real numbers. Is it a formula for the Vandermonde type determinant $V(x_1, \cdots,x_n)$ whose last column is $x_1^k,\ \cdots,\ x_n^k$, where $k \geq n$, instead of ...
3
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2answers
647 views

How to find k given determinant?

So I've got this matrix here, and need to solve for $k$ $$\text{det}\;\begin{pmatrix} 3 & 2 & -1 & 4 \\ 2 & k & 6 & 5 \\ -3& 2 & 1 & 0 \\ 6 & 4 & 2 & ...
4
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1answer
320 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
2
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1answer
102 views

Derivation of the $2\times 2$ determinant

Show that the determinant of any $2\times 2$ Matrix $A=\pmatrix {a& b\\c&d}$ is $ad-bc$ using the basic definitions. Proof: Perform the reduce row reduction algorithm on $A$ using the row ...