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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29 views

Why is the computation of the Jacobian determinant different for certain integrations?

I am used to computing the Jacobian, when, say, changing from x,y coordinates to u,v coordinates, as computing the determinant of the derivative matrix of $x_u$,$x_v$, $y_u$, $y_v$, i.e., ...
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0answers
38 views

Need clarification regarding a proof about the determinant of a block matrix

Let $A= (a_{ij}) \in M_n (F)$ be of the form \begin{align*} A = \begin{pmatrix} B & C \\ O & D \end{pmatrix}, \end{align*} where $B = (b_{ij}) \in M_r (F), D = (d_{ij}) \in M_s (F)$ and $C = ...
1
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0answers
22 views

Invertible matrices, permutations and leading principal minors

Given an invertible $\{-1,0,1\}$-matrix $A$ (its determinant is $\pm 1$), are there two permutation matrices $P$ and $Q$ such that all the leading principal minors (determinants of the top-left ...
2
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4answers
54 views

A determinant made of $n \times n$ determinants.

I came across this problem, in a recent exam. So I was given three matrices $$ A, B, C \in M_{n} (\mathbb{R}) $$ and that $$ 0 \in M_{n}(\mathbb{R}) $$ is the zero matrix. Then I was also given the ...
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1answer
46 views

Determinants and monic polynomials [duplicate]

I wish to show that $$ \det \begin{pmatrix} x & a & a & a\\ a & x & a & a\\ a & a & x & a\\ a & a & a & x \end{pmatrix}=(x-a)^3(x+3a).$$ Obviously, I ...
2
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0answers
50 views

Can I perform elementary line operations to $\det(A-\lambda I)$ like this?

I have a $4 \times 4$ matrix: $$A = \begin{pmatrix} 2 & 3 & 1 & 0 \\ 4 & -2 & 0 & -3\\ 8 & -1 & 2 & 1\\ 1 & 0 & 3 ...
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4answers
55 views

Parameter Matrix Determinant

$A=\begin{bmatrix} ...
6
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2answers
206 views

Determinant of a Certain Block Structured Positive Definite Matrix

PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED ...
2
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0answers
31 views

Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

$A$ is a invertible $n \times n$ matrix. Show: $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$ Can someone give me a hint on how to show that?
2
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1answer
68 views

Easiest way to calculate the determinant of this 4x4 matrix

I have this 4x4 matrix: $$A= \begin{pmatrix} 2 & 3 & 1 & 0 \\ 4 & -2 & 0 & -3\\ 8 & -1 & 2 & 1\\ 1 & 0 & 3 & ...
2
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2answers
53 views

Find the jacobian

I'm been struggling with the problem for a quite some time now. I need to find the jacobian for the following : $$u=x-y$$ $$v=xy$$ What I did : $$x=y+u\\x=\frac{v}{y}\\y=x-u\\y=\frac{v}{x}$$ ...
1
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0answers
38 views

An identity with determinant and trace of a matrix

How to prove the following identity: $$\det(A)=\frac{1}{d!}\sum_{\sigma\in S_d}\mathrm{sgn}(\sigma)\mathrm{Tr}_{\sigma}(A)$$ where $\mathrm{Tr}_{\sigma}(A)$ is defined as following if $\sigma$ is ...
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1answer
43 views

Determinant of an $n \times n$ matrix,problem

$ A_=\begin{pmatrix} ...
0
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0answers
29 views

Show a regular complex symmetric square matrix is reversible

Find following complex symmetric square matrix's canonical form under the congruent ( through complex square matrix ),where $i^2=-1$. $$\sum\limits_{1 \le k < l \le n} {\left( {k + il} ...
0
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1answer
23 views

How to check the determinant and rank of multiplied matrices?

Given $A \in\mathbb{R^{7\times8}}$, $B \in\mathbb{R^{8\times5}}$ and $C \in\mathbb{R^{5\times7}}$ How can one check whether $$det(ABC) = 0$$ is true? Given their spaces, the multiplications are ...
2
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2answers
30 views

Matrix roots of the characteristic equation

Let A be a matrix of $n \times n$ dimensions and $p( \lambda)= \det (A- \lambda I)$. Then $p(A)=0$ by Caylee-Hamilton. Are there any other matrices that satisfy the characteristic equation of A?
2
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0answers
31 views

Geometric Interpretation of Determinant of Transpose

Below are two well-known statements regarding the determinant function: When $A$ is a square matrix, $\det(A)$ is the signed volume of the parallelepiped whose edges are columns of $A$. When $A$ is ...
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1answer
31 views

Using a determinant to find the Cartesian equation for a plane from its parametric equations

This horribly unreadable webpage describes a method to find the Cartesian equation for a plane given its parametric equations. I'll try to type the method out here in a neater fashion: The ...
0
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2answers
36 views

Signing of a binary matrix to a totally unimodular matrix

I have the following binary matrix: \begin{pmatrix} 1& 1& 1& 0 \\ 0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ \end{pmatrix} Definition: Signing a matrix ...
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0answers
59 views

Find the Determinant Question

Find the determinant $$ \begin{vmatrix} \dfrac1{a_1+b_1} & \dfrac1{a_1+b_2} & \ldots & \dfrac1{a_1+b_n} \\ \dfrac1{a_2+b_1} & \dfrac1{a_2+b_2} & \ldots & \dfrac1{a_2+b_n} ...
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0answers
26 views

Why does this equality stand?

We have that $$\frac{\partial}{\partial{t}}J=\begin{vmatrix} \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & ...
2
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2answers
56 views

If $(I-A)(I+A)^{-1}$ is orthogonal then prove that A is skew symmetric.

Question from Determinants.Can't solve !
6
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2answers
73 views

Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis

I've been trying to solve this linear algebra problem: You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$. ...
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1answer
19 views

Computing determinant of the matrix $C$

Let $$C=\begin{bmatrix} 0 & 0 & \cdots &0 & -c_0 \\ 1 & 0 & \cdots & 0& -c_1 \\ 0& 1 & \cdots & 0& -c_2 \\ \vdots & \vdots & & & \\ 0 ...
3
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4answers
62 views

Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same ...
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1answer
31 views

Determinants using elementary row operations

Let matrix $A$ be defined as \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ ...
3
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2answers
101 views

Show determinant of $\left[\begin{matrix} A & 0 \\ C & D\end{matrix}\right] = \det{A}\cdot \det{D}$

Let $A \in \mathbb{R}^{n, n}$, $B \in \mathbb{R}^{n, m}$, $C \in \mathbb{R}^{m, n}$ and $D \in \mathbb{R}^{m, m}$ be matrices. Now, I have seen on Wikipedia the explanation of why determinant of ...
10
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1answer
154 views

Show that a matrix has positive determinant

For a natural number $i>0$, let $p_i$ be the $i$th prime number, that is, $p_1=2, p_2=3, p_3=5,...$. Show that for all $n$, the following matrix has positive determinant $$ \begin{pmatrix} ...
3
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2answers
251 views

Determinant of matrix with trigonometric functions

Find the determinant of the following matrix: $$\begin{pmatrix}\cos\left(a_{1}-b_{1}\right) & \cos\left(a_{1}-b_{2}\right) & \cos\left(a_{1}-b_{3}\right)\\ \cos\left(a_{2}-b_{1}\right) ...
2
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1answer
147 views

determinant of infinitely large matrix by decomposition

Read the too long didnt read version in bold before going into the finer detail. The overall point is that when I decompose this matrix to try and find its determinant I get an answer that doesn't ...
2
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1answer
39 views

Determinant of an Operator with No Eigenvalues

Suppose V is a real vector space. Suppose an operator on V, T, has no eigenvalues. Prove that det T $\gt 0$ I know that every operator on an odd dimensional real vector space has an eigenvalue and ...
2
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0answers
48 views

Linear Algebra - Determinant Properties

A = \begin{bmatrix} a & b & c \\[0.3em] d & e & f \\[0.3em] g & h & i \end{bmatrix} B = \begin{bmatrix} g & ...
0
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1answer
38 views

How do I find such matrices $X_{1},\ldots,X_{9} \in \mathrm{M}_{2}(\mathbb{Z}) $?

Is there someone who can give at a least an idea for solving this problem? Determine the matrices $ X_{1} , X_{2} , ..., X_{9} \in \mathrm{M}_{2}(\mathbb{Z})$ such that: $$(X_{1})^{4} + ...
1
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1answer
29 views

Linear System - Laplace - Determinant

Can somebody help me? I need to find the determinant of the related matrix with Laplace's method. What is the easiest way to find it? $x+y-z+w=1\\ x+2y+z-w=-1\\ y+2z-2w=-2\\ kx+3z=0$ Thank you for ...
1
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0answers
35 views

How to find the conjugate of a matrix

To find the adjoint of a matrix first we have to find the conjugate of matrix. for a 3X3matrix \begin{bmatrix} 1&-1& 1 \\ 1&2 & 2\\1&1&2 \end{bmatrix} some one explain me how ...
2
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4answers
40 views

How to compute determinant of $n$ dimensional matrix?

I have this example: $$\left|\begin{matrix} -1 & 2 & 2 & \cdots & 2\\ 2 & -1 & 2 & \cdots & 2\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 2 & 2 ...
6
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1answer
162 views

Evalute big determinant

Today in exam I tried to evaluate this determinant but failed, only somehow "guessed" the answer I got here. Now in home I've managed to find something intuitive, just want to know whether the ...
6
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1answer
145 views

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $A = \begin{pmatrix} ...
2
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1answer
39 views

Alexander polynomial of unknot without Fox calculus or infinite cyclic cover

As explained in Lickorish`s book "Introduction to knot theory", one can define the Conway-normalized version of the Alexander polynomial by the determinant of certain sum of Seifert matrix plus ...
4
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1answer
36 views

Determinant defined using multilinear alternating maps, and invertibility of linear endomorphisms

In Jeffrey Lee's differential geometry text on page 353 he defines the determinant in an interesting way using multilinear alternating maps: Suppose $V$ is an $n$-dimensional $k$-vector space over ...
0
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2answers
34 views

conditions for the value of a determinant to be zero

The theory states that the value of a determinant will be zero if it contains a row or column full of zero or if has two identical rows or two rows proportional to each other. similarly can we say ...
2
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1answer
34 views

Singular matrix with entries in a ring. [duplicate]

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
3
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3answers
78 views

If $A$ is a matrix, and $A^2=I$, then can I say that $|A|= \pm1$?

$A^2=I$ Take determinant on both sides: $$|A^2|= |I| $$ $$|A|^2= 1$$ $$|A| = +1 \text{ or } -1$$ Is this proof correct?
1
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1answer
24 views

The derivative of det(X'A) when X is a non-square matrix

For a non-square matrix $X$ of size $n \times p$ ($n>p$) and another non-square matrix $A$ of size $p \times n$, what is the derivative of $\det(X^TA)$ w.r.t. $X$? i.e., ...
4
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1answer
65 views

Determinant proof using its properties

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = ...
5
votes
1answer
63 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
4
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2answers
70 views

Determinant of matrices without expanding [duplicate]

Show that $$\begin{array}{|ccc|} -2a & a + b & c + a \\ a + b & -2b & b + c \\ c + a & c + b & -2c \end{array} = 4(a+b)(b+c)(c+a)\text{.}$$ I added the all rows but couldn't ...
4
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1answer
61 views

What operations can I do to simplify calculations of determinant?

My question is simple. Given an $n \times n$ matrix $A$, what operations can we do to the rows and columns of $A$ to make the calculation of its determinant easier? I know we can put it into row ...
1
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1answer
20 views

Calculating the determinant of an iterationmatrix

Let $C_\omega = (I-\omega D^{-1}L)^{-1}((1-\omega)I+\omega D^{-1}R)$ then $\det(C_\omega) = (1-\omega)^n$ (Where $C_\omega\in \mathbb{R}^{n\times n}$, $R$ is upper triangular, $L$ is lower ...
2
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3answers
137 views

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...