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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1answer
48 views

Inverse of sum of fractions

I'm interested in the inverse of a finite sum of fractions. eg: $$ \large{\frac{1}{\sum_{i=1}^{n} \frac{a_i}{b_i} }}$$ For $a_i, \ b_i \in \mathbf{R}$. Specifically, can this be expressed in terms ...
0
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1answer
31 views

How is the Jacobian linked to the determinant of a transformation?

I need to show that if $(X,Y,Z)^T = A(x,y,z)^T$ then $\dfrac{\partial(X,Y,Z)}{\partial(x,y,z)} = \det(A)$ I sort of understand the link between change in volume and Jacobians and determinants but ...
3
votes
2answers
66 views

Is this determinant identity correct?

For complex valued matrices $A,B$ where $B$ is invertible, does $$\det(I+B^{-1}AA^*)=\det(I+AA^*B^{-1})=\det(I+AB^{-1}A^*)=\det(I+A^*B^{-1}A)?$$ Here $A^*$ is the conjugate transform. I guess ...
3
votes
1answer
375 views

Prove that the determinant of $ A^{-1} = \frac{1}{det(A)} $- Linear Algebra

If I have a single matrix A that is non-singular, how can I prove the determinant of its inverse = $\frac{1}{\det(A)}$? Prove: $$ \det(\mathbf{A^{-1}}) = \frac{1}{\mathbf{\det(A)}} $$ I know that ...
2
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2answers
915 views

Q: The determinant of a matrix $A \in \mathbb{R}^{n \times n}$?

I really struggle with this problem, how do you calculate the determinant of matrix $A \in \mathbb{R}^{n \times n}$, whose expression is $$ \begin{pmatrix} 2 & 1& ...& 1\\ 1& ...
0
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2answers
615 views

Determining whether the system will have a nontrivial solution?

Say I have a 3x3 matrix (a1 = 3a2 - 2a3), Will they system Ax=b have a nontrivial solution? Is it non-singular? I realize nontrivial means an answer that is not a zero vector. It must be the ...
1
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2answers
44 views

The 8 vectors to be made non-collinear

Consider the set of $8$ vectors $V=\{ai+bj+ck:a,b,c \in \{-1,1\}\}$. How can I choose three non-collinear vectors from $V$? My try: Let there be three vectors \begin{align*} ...
0
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1answer
41 views

Can I use this formula with pseudo determinants instead of usual determinants?

Let $A$ be a matrix with $A^+$ Moore-Penrose inverse. Let also $Det()$ denote the pseudo-determinant of a matrix. Does the formula (which assumes the existence of $A^{-1}$) $$ det\left( ...
1
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2answers
106 views

Wronskian determinant and Linear dependence

I was trying to show that if functions f and g defined on interval I are linearly dependent then the Wronskian determinant is zero. Suppose f, g $\in$ I and f g are linearly dependent, then $\forall ...
4
votes
4answers
452 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
6
votes
5answers
296 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...
5
votes
2answers
411 views

Determinant (and invertibility) of generalized Vandermonde matrix

I have stumbled upon the following generalization of Vandermonde matrix when solving some problem in linear algebra related to Jordan normal form. Let us consider some number $\lambda$ and we assign ...
0
votes
1answer
408 views

Is ${v_1, v_2}$ a basis for $\mathbb{R}^3$ or $\mathbb{R}^2$?

Let $$v_1= \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix},\quad v_2 = \begin{bmatrix} -2 &\\ 7\\ -9 \end{bmatrix}$$ Will it be a basis for ...
1
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2answers
76 views

Matrix inverse exists even determinate is zero.

We know matrix inverse does not exist if det(matrix)=0. Now a 2*2 matrix with all entries $x$ has inverse as 2*2 matrix all entries $1/(4*x)$ . So what is the gap of understanding?
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4answers
706 views

Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
13
votes
3answers
240 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 ...
1
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0answers
29 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
2
votes
3answers
66 views

Finding General Formula of a Determinant

Let $A=(a_{ij})\in \mathbb{M}_n(\mathbb{R})$ be defined by $$ a_{ij} = \begin{cases} i, & \text{if } i+j=n+1 \\ 0, & \text{ otherwise} \end{cases} $$ Compute $\det (A)$ After ...
7
votes
2answers
177 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 ...
0
votes
3answers
71 views

Calculate determinant of $n \times n$ depending on n

My task is to figure out determinant of following matrix depending on $n$. I want to solve it without altering the rows! $$ A^{n,n} = \begin{vmatrix} 0 & & ... & 0 & -1\\ ...
0
votes
1answer
91 views

How to prove that determinant with permutation symbols

How to prove that $$\varepsilon_{ijk}a_{i\ell}a_{jm}a_{kn} = \det[a]\epsilon_{\ell mn}$$ I'm trying to solve this problem with permutation symbol but i can't solve it Help me,please. Thank you ...
1
vote
1answer
47 views

Defining determinants as product preserving functions

In most treatments I have seen, determinants of square matrices are either defined via Liebniz's formula, or as the unique function that is multilinear in each row, $0$ if two rows (or columns) are ...
1
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1answer
240 views

Alternating multilinear map and products

I was reviewing some school notes from many semesters ago and I came across a point which I wish to prove but can't. Let $F$ be a field (real or complex for example), and we say $\delta : ...
6
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0answers
206 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 ...
3
votes
0answers
52 views

How to prove that the determinant is the same no matter how you take it?

To find the determinant, pick a row and move along it creating minors and use the recursive definition of determinant. How do we know that the determinant will be the same no matter which row you ...
21
votes
3answers
581 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 $$\det(A+B)=\det B.$$
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0answers
41 views

Integral of a determinant of Jacobian depends on the boundary values only

Let $B$ be the closed unit ball in $\mathrm{R}^n$ with the 2-norm. Let $\phi : B \to \mathrm{R}^n$ be smooth such that $\det D \phi = 1$ on $\partial B$. Why is $\int_B \det D \phi = \int_B 1$? In ...
1
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3answers
92 views

Let $A=(a_{ij})$ be a $n \times n$ matrix where $a_{ij} = \max(i,j)$. Find the determinant of $A$.

Let $A=(a_{ij})$ be a $n \times n$ matrix where $a_{ij} = \max(i,j)$. Find the determinant of $A$. How to find the solution of this kind of problem?
4
votes
1answer
66 views

Special determinant (from Kostrikin's book).

Calculate below determinant by using product of determinants: $\left[\begin{array}{ccc}(a_o+b_0)^n & ... & (a_0+b_n)^n\\ \vdots & \ddots& \vdots\\(a_n+b_0)^n & ... & ...
3
votes
1answer
83 views

Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
4
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1answer
43 views

The determinent of a vector? Where those it come from and then is is useful and true?

The determinant of a vector $\vec u$ and $\vec v$ is: $$\operatorname{det}(\vec{u},\vec{v})=\Big|\begin{matrix}a & c \\ b & d \end{matrix}\Big|=a\times d-b\times c$$ But what is it really? ...
0
votes
1answer
49 views

What is wrong with my determinant calculation?

$$\left| {\begin{array}{*{20}{c}} 2 & 6 & 4 & 0 \\ 2 & 0 & 4 & 2 \\ 0 & 3 & 2 & 1 \\ 2 & 6 & 4 & 8 \\ \end{array}} \right| = \left| ...
1
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2answers
5k views

Set of Linear equation has no solution or unique solution or infinite solution?

For the system $$ \left\{ \begin{array}{rcrcrcr} x &+ &3y &- &z &= &-4 \\ 4x &- &y &+ &2z &= &3 \\ 2x &- &y &- &3z &= &1 ...
2
votes
1answer
48 views

Determinant of $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$

Calculate the determinant of the following matrix: $M \in M_{2n}(\mathbb{C})$ such that $$M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$$ I find that that $\det M = 2^n$ is that ...
4
votes
2answers
125 views

Calculate the determinant of given matrix

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
2
votes
1answer
309 views

Determinant of block matrix with certain properties

So I have the following 2N $\times$ 2N block matrix $H=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ where each block in an N$\times$N matrix. Each block have the following ...
0
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1answer
50 views

Difficulty proving formula containing the adjugate and determinant of a matrix

This is what I need to prove: You have an invertible matrix $A \in M_3(\Bbb R^3)$. Prove that $\operatorname{adj}(\operatorname{adj}(A))=\det{(A)}^{n-2}A$ The proof goes as follows: ...
3
votes
1answer
75 views

Prove $\frac{c_n(a_1,…,a_n)}{c_{n-1}(a_2,…,a_n)}=a_1 + \cfrac{1}{a_2 + \cfrac{1}{\ddots + \cfrac{1}{a_{n-1}+\frac{1}{a_n}}}}$

For $n>0$ and $a_1,...,a_n \in K$ let $c_n(a_1,...,a_n)$ be the determinant of the matrix $$ \begin{pmatrix} a_1 & 1 & 0 & \cdots & 0 \\ -1 & a_2 & \ddots & ...
1
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1answer
54 views

Determinant of parametric function and $0!1!2!…n!$

As answer to this question, I trued to calculate the wronskian of: $$\left| \begin{array}{ccc} e^x & e^{2x} & ... & e^{nx}\\ e^x & 2e^{2x} & ...& ne^{nx} \\ e^x & 4e^{2x} ...
1
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2answers
298 views

Gram Determinant equals volume?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly m =< n. In general, for computational purposes, what I have managed to do ...
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2answers
69 views

If $AB=-I_n$, then prove that $det(I_n+BA)=2^n$

Given two matrices $A,B\in \mathbb M_n(\mathbb R)$ and $AB=-I_n$, prove that: $$det(I_n+BA)=2^n$$ We know that: $2^n=det(2\cdot I_n)=det(I_n+I_n)$ and $I_n+BA=-(-I_n-BA)=-(AB+BA)$. How can I get ...
4
votes
2answers
98 views

Recursive determinant of given matrix in $\mathbb{R}^{n\times n}$

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
2
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1answer
209 views

determinant recursive formula of a specific matrix

For a field $K, n \in \mathbb{N}_{>0}$ and $\lambda \in K$ let $A_{n, \lambda} \in \textrm{Mat} (n,K) $ be the following matrix with entries $\lambda$ on the diagonal, $-1$ on both minor diagonals ...
3
votes
3answers
109 views

Determinant of a special $n\times n$ matrix [duplicate]

Compute the determinant of the nun matrix: $$ \begin{pmatrix} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 ...
0
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1answer
79 views

Prove that these determinants are equal

Without calculating the determinant, prove: $ \begin{vmatrix} 0 & x & y & z \\ x & 0 & z & y \\ y & z & 0 & x \\ z & y & x & 0 \\ \end{vmatrix} = ...
1
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1answer
47 views

Proving a determinant inequality

Let $A$ be a square matrix in $M_n(\mathbb R)$. Prove that: $$det(A^2+I_n) \ge 0$$ I wrote $A^2+I_n=A^2 I_n+I_n=I_n(A^2+1)$: $$det(I_n)\cdot det(A^2+1)=det(A^2+1)$$ How can I prove that is $\ge 0$ ...
2
votes
1answer
260 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
3
votes
2answers
3k views

Relation between determinant and matrix rank

Let $A$ a square matrix with the size of $n \times n$. I know that if the rank of the matrix is $\lt$ $n$, then there must be a "zeroes-line", therefore $det(A) = 0$. What about $rank(A)=n$? Why ...
7
votes
5answers
325 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 ...
1
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0answers
77 views

Problem about the determinant of a random matrix

I am being haunted about this problem on the value of the determinent of this Random Matrix ever since it came into my mind last week.The problem goes like this: Suppose $A$ is a square matrix of ...