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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2
votes
1answer
19 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
8
votes
1answer
412 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
2
votes
4answers
35 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
1
vote
1answer
32 views

Find isomorphism between $S_3$ and $GL_2(F_2)$. [duplicate]

Find isomorphism between $S_3$ and $GL_2(F_2)$. proof: Let $A = \begin{pmatrix} a& b\\ c & d \end{pmatrix}$. Where $\det (A) \neq 0$. And recall $S_3$ is the permutation group with ...
2
votes
0answers
47 views
+50

What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
1
vote
2answers
32 views

determinant of a vector times vector transpose

I have a vector $x$ of dimension $N \times 1$ and let's say I create a matrix $S = x x'$ which a matrix of dimension $N \times N$. If I calculate the determinant of $S$, I get it as $0$. Is this a ...
2
votes
0answers
23 views

Problem with determinant

Let $A\in\mathbb{C}^{3\times 3}$ and $x,y\in\mathbb{C}^3$. Prove that $det\left(I-\frac{xy^*A}{1+y^*Ax}\right)=\frac{1}{1+y^*Ax}$ How can I prove this?
2
votes
6answers
59 views

For $n\times n$ matrices, is it true that $AB=CD\implies AEB=CED$?

If $A,B,C,D,E$ are $n\times n$ matrices, does $AB=CD$ imply $AEB=CED$? I only know that $AB=CD \implies ABE=CDE$, but I don't see how you can sandwhich $E$ within it. Also, if $AB=CD=0$, does ...
3
votes
2answers
2k views

Matrix determinant using Laplace method

I have the following matrix of order four for which I have calculated the determinant using Laplace's method. $$ \begin{bmatrix} 2 & 1 & 3 & 1 \\ 4 & 3 & 1 & 4 \\ -1 ...
2
votes
1answer
19 views

Equality of determinants for a specific collection of square matrices of size $n=2^m$

My investigations have led me to a question that I am convinced is true. I need to show that, for a given $m$, a certain collection of square $n=2^m$ matrices have the same determinant. In dimension ...
3
votes
0answers
37 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
2
votes
1answer
20 views

Log-determinant ordering for sum of positive definite symmetric matrices

If, for real positive definite symmetric $A, B, C$, $$\log\det (A+B) \geq \log\det(A+C)$$ then can it be said that $$\log\det(B) \geq \log\det(C)?$$ NOTE: A crude form of the reverse is certainly ...
7
votes
1answer
68 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 ...
1
vote
0answers
10 views

$det(I+A(\epsilon))$ where $A$ is an infinite matrix and not trace class!

Assume that $A$ is an infinite matrix and it's a function of the parameter $\epsilon$. I would like to find $\epsilon$ so that the $det(I+A(\epsilon))=0$. I know if $A$ was a trace class I could use ...
-5
votes
0answers
30 views

show that .i write degree in words .becze there is no optn of degree [closed]

Sin 10 degree -cos 10 degree = 1 Sin 80 degree cos 80 degree Show that this diterminant is equal to 1
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
9
votes
1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
2
votes
0answers
25 views

Evaluating determinant [duplicate]

Let $\{\alpha_{i}\}_{i=1}^{n}$ be distinct numbers. What is the determinant of the $n$ by $n$ matrix \begin{gather} \begin{pmatrix} \alpha_{1}^{n-1} & \alpha_{1}^{n-2} & \cdots & 1 \\ ...
1
vote
1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
2
votes
2answers
293 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
149 views

Proof $\det(AB)=\det(A)\det(B)$

I have read the following proof , in here: Why can we go from the first line to the second one? why $Det(E^k)\cdot Det(E^m)=Det(E^k\cdot E^m)$? is it because $\det(E^k)\in \mathbb{F}$ for all ...
14
votes
1answer
167 views

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 ...
-1
votes
1answer
69 views

Simplifying a sum of products related to Vandermonde determinant

How to show this equality? $$ 1=(-1)^n\sum_{k=0}^n\frac{x_k^n}{\prod_{\substack{l=0 \\ l \neq k}}^n(x_l-x_k)} $$ This is part of a proof to show the value of the determinant of the Vandermonde matrix ...
1
vote
3answers
69 views

Find the values of 'a' in a $4\times 4$ matrix(A) when the determinant is less than 2012

The matrix is $A \ =\begin{pmatrix} 7 & 1 & 3 & -2\\ -2 & 1 & -12 & -1 \\ 1 & 16 & -4 & a \\ ...
0
votes
1answer
32 views

If a NxN matrix has two identical columns will its determinant be zero?

I am currently doing a practice final for a Linear Algebra Class. In it I am given the following statement and asked to determine whether it is true or false. "If det(A) = 0, then two rows or two ...
1
vote
3answers
45 views

Is $\det(c I_n - A^T) = \det(c I_n - A)$? How to prove it?

Problem: Are the following assertions true or false? Prove or give a counterexample: 1) If $A$ is an $(n \times n)$-matrix, then for every $c \in \mathbb{R}$ we have $\det(c I_n - A) = c^n - ...
1
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0answers
22 views

Derivative of a determinant with respect to a matrix

Can someone tell me the derivative of the following determinant ($\Psi\in\mathbb{R}^{p\times p}$, $Z\in\mathbb{R}^{p\times q}$, $\alpha\in\mathbb{R}^q$) $\frac{\partial}{\partial \Psi} ...
2
votes
2answers
104 views

Prove $\det(A+BC)=\det(A+CB)$ if $AB=BA$ [closed]

Let $A$, $B$ and $C$ be three endomorphisms of a finite-dimensional vector space such that $AB=BA$. Prove that $$\det\left(A+BC\right)=\det\left(A+CB\right)$$
1
vote
2answers
30 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
-4
votes
4answers
90 views

Determinants with arithmetic progressions as columns [closed]

Prove that determinants of the following form all vanish: $$\det \begin{bmatrix} x-3 & x-4 & x-a \\ x-2 & x-3 & x-b \\ x-1 & x-2 & x-c\end{bmatrix} = 0$$ Here $a$, $b$, $c$ are ...
0
votes
0answers
27 views

Algorithm for finding the value of determinant

Okay I am writing to write a program which computes the determinant of a matrix. So is there an algorithm that allows you to do that ? Are there any other ways of finding the determinant value other ...
1
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3answers
56 views

Why is determinant a multilinear function?

I am trying to understand (intuitive explanation will be fine) why determinant is a multilinear function and therefore to learn how elementary row operation affect the determinant. I understand that ...
1
vote
1answer
26 views

Alternating multilinear function satisfies $f(A)=\det(A)f(Id)$

I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$." The determinant is defined as the ...
7
votes
1answer
415 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: ...
2
votes
1answer
40 views

Prove that det($A$) is non-zero iff $A$ is row equivalent to the $n\times n$ identity matrix

$A$ is an $n\times n$ matrix. Now if the row-reduced echelon form for this $A$ is $E$ then after all the row operations we have $\det(A)=M\det(E)$ where $M$ is a non-zero ...
0
votes
1answer
18 views

Integration of exponential matrix and determinant?

Is it possible to prove $$\int \exp\{-\frac{1}{2}(\beta-\hat\beta)^T(X^TH^{-1}X)(\beta-\hat\beta)\}\text{d}\beta=\{\det(X^TH^{-1}X)\}^{-1/2},$$ where $\hat\beta,X,H$ are all known? What additional ...
27
votes
1answer
856 views

Prove the determinant of this matrix

We have an $n\times n$ square matrix $\left(a_{i,j}\right)_{1\leq i\leq n, \ 1\leq j\leq n}$ such that all elements on main diagonal are zero, whereas the other elements are defined as follows: ...
1
vote
1answer
54 views

What is $\mid\text{det}(A,G)\mid$?

I am reading an old paper dated back in 70', where I encounter this $$\mid\text{det}(A,G)\mid=(\text{det}\{(A,G)'(A,G)\})^{\frac{1}{2}}.$$ We compute the determinant of a single matrix, don't we? ...
1
vote
2answers
344 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 ...
5
votes
1answer
41 views

How to prove that this matrix is total unimodular

This matrix is total unimodular (tested by a computer program). ...
2
votes
2answers
303 views

Finding determinants by inspection?

I'm supposed to "use properties of determinants to evaluate the determinant by inspection" on this matrix: $$\begin{bmatrix} 4 & 1& 3\\ -2 & 0 &-2 \\ 5 & 4 & ...
0
votes
1answer
13 views

relation between special linear group and special orthogonal group

What is the difference between special linear group and special orthogonal group ? The special linear group is the set of endomorphisms with determinant $1$. On the other hand, the special ...
4
votes
1answer
412 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
votes
1answer
54 views

Find $a$ in the following matrix

I have the following question : matrix $A$ isn't diagonalizable while $a \in R$ $$A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}$$ Find $a$. I ...
3
votes
2answers
63 views

Determinant of $ n \times n$ matrix and its characteristic polynomial.

Suppose, $M_4, M_5,..M_n$ is as follows then determinant and characteristic polynomial of $M_n$. $M_4=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 ...
1
vote
1answer
42 views

Finding the volume of the set of all $x \in \mathbb R^4$ satisfying $x^t A x \leq 1$ for a symmetric matrix $A$

Find the volume in $\mathbb R^4$ of the set of $x$ with $x^tAx \le 1$. You may use the fact that the volume in $\mathbb R^4$ of the set of $x$ with $|x|^2 = x^tx\le 1$ is $\frac{\pi^2}{2}$. My ...
0
votes
1answer
26 views

Convexity of Determinant of linear combination

Is it possible to show that the following is a convex function in $x$? $f(x)=\det(\sum_i x_i A_i)$ $A_i$ are real symmetric, positive definite matrices. Minkowski's inequality doesn't seem to do ...
2
votes
1answer
48 views

Why adjugate of $A$ is non singular, when $A$ is non singular?

Let $A$ be a non-singular square matrix. We know that $A \cdot \operatorname{adj}A = \det A \cdot I$. This implies that $\det\left(\operatorname{adj} A\right) = \left(\det A\right)^{n-1}$. Hence ...
1
vote
2answers
659 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
1
vote
0answers
21 views

Proof of determinant formula and coprime polynomial

Problem: Let $p(z)=p_o+p_1z+...+p_{n-1}z^{n-1}$ be a polynomial of maximum degree $n-1$. Show that $p(z)$ and $z^n-1$ are coprime if and only if $$\begin{vmatrix} p_0 & p_{n-1} & ... & ...