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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15
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3answers
385 views

Prove that $\det(A+B)=\det B$

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 $$\det(A+B)=\det B.$$
1
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0answers
27 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
vote
3answers
65 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?
0
votes
0answers
51 views

Determinant of almost symmetric matrix

I've got another problem with calculating determinant: Calculate determinant of matrix $M$, where $M$ is matrix such that elements are symmetric to diagonal $a_{1,n}, a_{2,n-1},...,a_{n,1}$. I was ...
4
votes
1answer
47 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
0answers
44 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
votes
1answer
38 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
40 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| ...
0
votes
1answer
517 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
36 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
89 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
101 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
votes
1answer
28 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
52 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
vote
1answer
35 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
votes
2answers
61 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
73 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
votes
1answer
48 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 ...
2
votes
3answers
65 views

Determinant of a special $n\times n$ matrix

Compute the determinant of the nun matrix: $$ \begin{pmatrix} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 ...
0
votes
1answer
62 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
vote
1answer
42 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
111 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 ...
1
vote
2answers
248 views

the relation between determinant and a 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
169 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
46 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 ...
4
votes
2answers
183 views

Another way to look at determinants

Let $A$ be an $n\times n$ non-singular matrix with real entries. How can I prove the following equation? Any references would be helpful. $$ \det(A) = \frac 1{n!} \left| ...
4
votes
2answers
158 views

Determinant of a matrix with generalized binomial coefficients

Let $$ A= \begin{bmatrix}\binom{-1/2}{1}&\binom{-1/2}{0}&0&0&...&0\\ \binom{-1/2}{2}&\binom{-1/2}{1}&\binom{-1/2}{0}&0&&...\\...&&&\binom{-1/2}{0}\\ ...
4
votes
2answers
85 views

$T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
2
votes
1answer
75 views

Differentiating the determinant of the Jacobian of a diffeomorphism (don't understand a proof)

For each $t$, let $A_t:\Omega_0 \to \Omega_t$ be a bi-Lipschitz map between open sets in $\mathbb{R}^n$. The map is also invertible. It satisfies $$\frac{d}{dt}A_t(y) = w(A_t(y),t)$$ where $w$ is a ...
1
vote
2answers
148 views

When this hold: $\det(AA^{T})=0?$

I know if A is a column vector, the equality holds. Any comments or suggestions would be greatly appreciated.
1
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2answers
93 views

Proof determinant of transpose Vandermonde matrix is $\prod_{1\le i\lt j\le n}(\alpha_i-\alpha_j)$

the below is a transpose Vandermonde matrix determinant equality. I have seen a lot of proofs of its determinant being $=\prod_{1\le i\lt j\le n}(\alpha_j-\alpha_i)$, but this ones indices are ...
0
votes
2answers
72 views

Calculate determinant of matrix

Calculate the determinant of this matrix for $a, a_0,...,a_{n-1} \in K$ over any field $K$ $$ \begin{pmatrix} a & 0 & \cdots & 0 & -a_0 \\ -1 & \ddots & \ddots & \vdots ...
3
votes
1answer
118 views

Cramer's rule: Geometric Interpretation

I have a question concerning Cramer's rule: Let $A$ be a matrix and $A \cdot \vec x = \vec b$ a lineare equation. $A_i$ is the matrix $A$ where the i'th column is replaced by $\vec b$ if $det(A) ...
2
votes
2answers
92 views

Divisibility of determinants and matrix rows

I have stumbled upon an exercise that is giving me nighmares. I've found that it is quite common in older exercise books, but I haven't even heard about it in class or seen it in any lecture in all my ...
3
votes
2answers
70 views

how to compute the determinant of a linear map

Let $V$ be the vector space of $m\times n$ matrices over a field $F$. Fix an $m\times m$ matrix $A$ and an $n\times n$ matrix $C$, and consider the map $\phi: V\longrightarrow V$ defined by ...
1
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0answers
80 views

Special matrices with determinant 0

Define a quadratic matrix A with n rows and n columns by filling it with consecutive primes, starting with some prime p. The object is, to find the least starting prime p, such that A has determinant ...
0
votes
2answers
72 views

Invariant determinant in change of basis

So I've seen the proof for why the determinant of a transformation $T$ is the same under a change of basis, but I must have some basic misconception about this since I can't figure out what's wrong ...
0
votes
1answer
47 views

Determinant inequality for trace class operator

Let $A$ be a trace class operator on a Hilbert space. I wonder if there is an estimate of the form $$ |\log \det (I + A)| \le C\|A\|_1, $$ for some constant $C$, where the norm on the right is the ...
0
votes
1answer
76 views

Compute the determinant of the matrix and find a basis of its column space

The given matrix is $\begin{pmatrix} 1 & 1 & 0 & 0\\ -1 & 0 & 1 & 0\\ 1 & 3 & 4 & 1\\ 1 & 2 & 1 & 0\end{pmatrix}$ To obtain the determinant, I use ...
-1
votes
5answers
95 views

Definition of determinant [closed]

When determinant is a funtion from the set of all nxn matrices to a scalar, how can I define the definition of determinant? What characterizes the determinant function?
2
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1answer
55 views

determinant identity for invertible finite rank operators

I am currently reading a paper where the following identity, valid for an invertible finite - rank operator $T \colon \mathscr{H} \to \mathscr{H}$ on a separable Hilbert space, is given: $$ \log \det ...
0
votes
4answers
98 views

big determinant calculation

I have found this exercise in a book, and having troubles solving it: How to calculate this determinant? $$\det\begin{pmatrix} 5 & 6 & 0 & 0 & 0 & \cdots & 0 \\ 4 ...
2
votes
1answer
136 views

Check if matrix determinant is zero

What's the simplest way to check if a NxN Matrix determinant is zero ? Using Gauss Jordan to calculate the determinant first is to complicated (took N^3 calculation), is there any way to know it in at ...
0
votes
3answers
110 views

Compute the determinant $\begin{vmatrix} \sin x & \cos x \\ -\sin y & \cos y \end{vmatrix}$ using the Sarrus' rule

I have an exercise that asks to compute the determinant using the rule Sarrus $\begin{vmatrix} \sin x & \cos x \\ -\sin y & \cos y \end{vmatrix}$ The rule is not just for Sarrus $3\times 3$ ...
2
votes
2answers
102 views

How to calculate the determinant of this matrix $A=\begin{bmatrix} \sin x & \cos^2x & 1 \\ \sin x & \cos x & 0 \\ \sin x & 1 & 1 \end{bmatrix}$

How to calculate the determinant of this matrix $A=\begin{bmatrix} \sin x & \cos^2x & 1 \\ \sin x & \cos x & 0 \\ \sin x & 1 & 1 \end{bmatrix}$ ...
0
votes
2answers
54 views

$A$ is an $n\times n$ matrix and $A^2 = A$, then what are the possible values of $|A|$?

If $A$ is an $n\times n$ matrix and $A^2$ = A, then what are the possible values of |A|?
2
votes
1answer
49 views

Finding Jordanizing matrix

Let $$A=\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ And I found B, A's Jordan form to be: $$B=\left(\begin{matrix}0&1&0 \\ 0&0&0\\ ...
2
votes
0answers
34 views

The trace of a matrix is the sum of its eigenvalues [duplicate]

If $A$ is a complex square matrix, I need to prove that the trace of $A$ is the sum of its eigenvalues. I've already proved that, if $p(x)$ is the characteristic polynomial, then ...
1
vote
2answers
70 views

About eigenvalues and complex matrix

If $A$ is a square complex matrix with $n$ rows, prove that the constant term of the characteristic polynomial is equal to $(-1)^ndet(A)$ and that the coefficient of degree $n-1$ is equal to $-Tr(A)$ ...
1
vote
1answer
80 views

How to calculate pseudo-determinant for implementing Naive-Bayes

(People who followed Bayesian tag, please read the third paragraph) Problem: I need to calculate pseudo-determinant of a matrix (preferably in MATLAB, but no ...