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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33 views

Is there any easiest way to find the determinant? [duplicate]

Suppose, $ M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1} & -x & a_3&a_4&\cdots &a_n\\ a_1&a_{2} & -x ...
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71 views

Low-degree “determinant” for non-square matrices?

Consider a matrix $A\in \mathbb R^{n\times n}$ of indeterminates. The determinant of $A$ is a degree $n$ polynomial in the $n^2$ entries satisfying $\det A\ne0\iff A$ is nonsingular. What about when ...
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27 views

Show that the determinant is the Wronskian

Prove that the determinant of the following system $(\star)$ is the Wronskian. $$(\star) \begin{pmatrix} y_1(s) & -y_2(s)\\ -y_1'(s) & y_2'(s) \end{pmatrix} \begin{pmatrix} c_1(s)\\ c_2(s) ...
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1answer
52 views

Finding Eigenvalues of a 3x3 Matrix (7.12-17)

Please check my work in finding eigenvalues for the following problem. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons. For reference ...
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87 views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
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2answers
31 views

what condition of A makes transpose(A)*A nonsingular?

What contidion of A makes $$A^TA$$ nonsingular? If so, that is $$A^TA$$ is non-singular than a unique solution exists.
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9 views

An algebraic equation system and the Jacobi determinant as test for its solvability

I am trying to verify a result in a text that I am currently reading. The context is in algebra and combinatorics. However the result is obtained by using a bit of vector calculus much to my suprise. ...
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1answer
46 views

Calculating difference between two matrices to a scalar

I want to calculate the difference between two n x n matrices to a scaler. That measure should give the idea about the physical location of each values too. For example if I name some operation with ...
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2answers
21 views

What does the homogeneous system of equations represent under certain conditions?

Consider the following linear equations $ax+by+cz=0,bx+cy+az=0,cx+ay+bz=0$ 1) $a+b+c \neq o$ and $a^2+b^2+c^2=ab+bc+ca$ 2) $a+b+c \neq o$ and $a^2+b^2+c^2 \neq ab+bc+ca$ 3) ...
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3answers
65 views

To evaluate the given determinant

Question: Evaluate the determinant $\left| \begin{array}{cc} b^2c^2 & bc & b+c \\ c^2a^2 & ca & c+a \\ a^2b^2 & ab & a+b \\ \end{array} \right|$ My answer: $\left| ...
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38 views

Calculate Pfaffian of a special 2x2 block matrix

I have a $2\times2$ matrix $$ M= \begin{pmatrix} A & -1\\ 1 & B \\ \end{pmatrix}. $$ Here $A$ and $B$ are skew matrix, the matrix dimension is $L$. Is there a quick way to calculate the ...
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1answer
24 views

About a step in the proof about determinant of adjugate matrix

I was trying to understand this answer which is about $\det(\operatorname{adj}(A))$. I came across a step: |A adj(A)|=|(|A|I)|. The next step was: |A||adj(A)|=|A|^n * I. I didn't get how |(|A|I)|= ...
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0answers
42 views

How to calculate log determinant of a function of matrices

I am new to matrix calculus. My question is simply how to calculate $\frac{\partial \log(\det(A+bX))}{\partial b}$ where A and X are n by n matrices and b is a scalar. I was trying to use chain rule, ...
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3answers
88 views

If all entries of matrix $X$ are the same, then $\det (A+X)\det (A-X) \leq \det (A^2)$

I want to prove that $\det (A+X)\det (A-X) \leq \det (A^2)$ where $X $ is a matrix whose $n^2$ entries are all the same. I tried to write down the expressions involved but that didn't help me prove ...
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0answers
55 views

Determinant of a function

I was thinking about matrices and then why arent there matrices with uncountable many values? (Probably this conecpt already exists for a very long time, but i don't know it) Assume there are ...
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1answer
94 views

Multilinear and alternating property of $\det(f)$ where $f$ is an endomorphism

Everybody knows the determinant of a matrix $A\in k^{n\times n}$ ($k$ a commutative ring) and everybody knows that the determinant of $A$ is an alternating multilinear map in the columns aswell as in ...
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20 views

Fundamental system of solutions

Theorem Let $x_i^{(k)}(t), i, k=1, \dots, n$ be a fundamental system of solutions of $x'=Ax$. Then any solution of this system can be written as a linear combination of $x_i^{(k)}(t), i,k=1, \dots, ...
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1answer
25 views

Check Vector3 points on one line using a Matrix

I know that for 3 Vector2 points (say points a, b, c) the determinant of the following ...
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1answer
30 views

Minimal polynomial of a $4\times4$ matrix [closed]

I just need to see an example of a non-diagonalizable $4\times4$ matrix over $\mathbb{R}$ whose minimal polynomial is the same as its characteristic polynomial. I saw the question elsewhere and ...
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1answer
36 views

How do we deduce that it is the zero function?

Theorem If the Wronskian of $x^{(1)}(t), \dots, x^{(n)}(t)$, that are solutions of $x'=Ax$ on an interval, gets zero at some point $t=t_0$ of the interval, then $x^{(1)}, \dots, x^{(n)}$ are linearly ...
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2answers
130 views

Does this extreme huge size determinant converge?

Introduction: One day I calculated the value of determinant which is like Hilbert matrix $H_{n}^{p} \in \bf{R}^{\it{n \times n}}$using my computer. The determinant is defined below. $$ ...
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2answers
35 views

Determinant of complex matrix with almost constant lines

Let $0\neq c\in\mathbb{C}$. Take the matrix $$A_C=\begin{pmatrix} n&c&\dots&c&c \\ c&n&c &\dots & c\\ c &c & n &c &\dots\\ \vdots ...
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1answer
25 views

Transform the determinant to upper triangular form

I have difficulty transforming this determinant $$\begin{vmatrix} x&x & x &\dots & x& x\\ a_1 &x &x &\dots &x& x\\ 0& a_2& x&\dots &x& x\\ ...
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$AB - BA$ is invertible and $A^2 + B^2 = AB \implies 3 | n$ [closed]

Given $A,B \in \mathbb M_n (\mathbb R)$ and that $A^2 + B^2 = AB$ $AB - BA$ has inverse Prove that $3 \mid n$.
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2answers
574 views

Is there an easy way to compute this determinant

\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & a \\ 0 & 1 & 0 & 0 & a & 0 \\0 & 0 & 1 & a & 0 & a \\0 & 0 & a & 1 & 0 & a \\0 & ...
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0answers
22 views

Having trouble to find the value of the following determinant

I came across the following problem from the book Higher Algebra(by barnard and Child) that says: Prove that $\,\,\begin{vmatrix} bc &bc'+b'c &b'c' \\ ca& ca'+c'a &c'a' \\ ...
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1answer
27 views

Prove a Block Matrix is Positive-Definite Given the elements

Suppose I have a non-negative, symmetric $(n+1)\times(n+1)$ block matrix $$ M = \begin{bmatrix} A & B \\ B^T & 1 \end{bmatrix} $$ where $A$ is an $n\times n$ positive-definite matrix and $B$ ...
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2answers
33 views

Computing the determinant of a matrix which splits up into unequal blocks

Suppose I have a matrix of the following form: $M = \begin{bmatrix} A & \mathbb{v} \\ \mathbb{v}^T & a\\ \end{bmatrix}$ where $A$ is an $n \times n$ matrix, $\mathbb{v}$ is an $n ...
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1answer
45 views

Determinant of nth order

I want to solve the following determinant: $D_n= \begin{vmatrix} a_n & a_{n-1} & \cdots & a_2 & x\\ a_n & a_{n-1} & \cdots & x & a_1\\ a_n ...
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1answer
27 views

Logarithm of complex matrix

For invertible matrix $A$, we have $\log(\det A) = \mathrm{tr}(\log A)$ due to a corollary of Jacobi's formula. What if we had the argument $iA$ instead? Would the above relation still hold? Edit: ...
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1answer
22 views

How to prove the value of a “negative” matrix

So I came across the formula that $\det(-A)=(-1)^n \cdot det(A)$, where $n$ is the number of columns/rows of A. I know how you get the formula by Laplace's formula and only described in words somehow, ...
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0answers
30 views

Proove that $\det(L_B) = \big(\det(B)\big)^n$, $\det(R_B) = \big(\det(B)\big)^n$. [duplicate]

Let $B \in M_{n×n}(\Bbb F)$. Define the functions $L_B$ and $R_B$ by: $L_{B}(A) = BA$ and $R_{B}(A) = AB$. Prove that $\det(L_B) = \big(\det(B)\big)^n$, $\det(R_B) = \big(\det(B)\big)^n$. $L_B$ and ...
2
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2answers
34 views

Show that determinant is divisible by f(x) [closed]

Let $\alpha$ be a repeated root of the quadratic equation $f(x)=0$ and $A(x),B(x),C(x) $be polynomials of degree 3,4 and 5 respectively.Then show that \begin{vmatrix} A(x) & B(x) & C(x) \\ ...
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1answer
47 views

Theoretical question about rank and invertibility of a block matrix,

Let A and B be real matrices, A is symmetric, and B has at least as many columns as rows. $$ C= \begin{bmatrix} A & B^t \\ B & 0 \\ \end{bmatrix} $$ a) Prove ...
2
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2answers
54 views

Find the inverse and determinant of A=(aI +T),

where is $a\ne 0$, $T$ has rank-one and zero trace. I just verified that a rank-one matrix has at most one non-zero eigenvalue. Now since T is of rank-one and has zero trace, that means all of its ...
6
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1answer
126 views

Determinant of an unknown matrix.

Let $x, y$ be two real variables. If $A$ is any $n\times n$ matrix with all entries in the set $\{x,y\}$ then prove that \begin{equation} \det A = (x-y)^{n-1}(Px + (-1)^{n-1}Qy) \end{equation} where ...
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1answer
82 views

Matrices - One problem with two optional solutions, don't know which one is correct

I have the following question: Let $A$ be a $3\times 3$ matrix such that $|\text{adj}(3A)|=3$. Find $|A|$. I solved the question in two different methods, but one method gave a solution which ...
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0answers
31 views

Prerequisites for Blyth's Module theory - an approach to linear algebra

I would like to know from people who have read this book if it can be tackled without prior exposure to linear algebra. The author claims in the introduction that "algebraic prerequisiste is the ...
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1answer
64 views

For what values of $x$ is $\det\begin{bmatrix} x & 5\\ 7 & 10 \end{bmatrix} = 30$?

If I have a matrix of the form $$ A = \begin{bmatrix} 4a & b\\ 4b & a \end{bmatrix}, $$ then $$ \left | A \right | = 4a^{2} - 4b^{2} = 4\left( a^{2} - b^{2} \right) = 4 \left( a + b \right) ...
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4answers
73 views

Prove that the determinant is $0$ by expressing as a product

I need to prove that the determinant $$\begin{vmatrix} my+nz & mq-nr & mb+nc \\ kz-mx & kr-mp & kb-ma \\ nx+ky & np+kq & na+kb \end{vmatrix}=0$$ In my book it is given ...
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1answer
20 views

Proof based problem related to non-trivial solution of a linear equation system

If the system of linear equations $$a(y+z)-x=0$$ $$b(z+x)-y=0$$$$c(x+y)-z=0$$ has a non-trivial solution $(a,b,c \neq -1)$,then show that $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}=2$$ ...
2
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0answers
37 views

Looking for a proof of a known theorem involving integral quadratic forms

Let $n$ be a positive integer and let $Q$ be an integral quadratic form in $n$ variables. Let $M$ be the symmetric "two's in" matrix associated with $Q$ so that $Q$ can be expressed as the $1 \times ...
3
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0answers
65 views

Help me to prove the determinant of given matrix.

Suppose, $ M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1}+x & -x-a_2 & 0&0&\cdots &0\\ a_1+x&0 & -x-a_3 ...
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1answer
72 views

Proof of a determinant expansion

This is equivalent to a result in Prasolov's book on linear algebra whose proof is not clear to me. I need help in understanding why the result is true. Let $x_1,x_2,\dots,x_n$ be row vectors in ...
2
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1answer
48 views

The determinant of a certain matrix

How to compute the following determinant? $$\left| \begin{matrix} 1 & x_0 & x_0^2 & \ldots & x_0^n \\ 1 & x_1 & x_1^2 & \ldots & x_1^n \\ \vdots & \vdots & ...
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1answer
25 views

What does it mean for vectors of a matrix to be linearly dependent?

I'm studying matrices and the implications of the determinant being $0$. I've read that if the determinant of a transformation matrix is $0$, then the vectors in the rows or columns are "linearly ...
2
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1answer
30 views

Is every “weakly square” matrix either a $0$ matrix, or a square matrix?

Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly, every square matrix is weakly square, and every zero matrix is weakly square. Question. Are ...
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1answer
82 views

Determinant $n\times n$ problem

$$ D_n = \left| \begin{matrix} n & -1 & -3 & 0 & 0 & \cdots & 0 & 0 & 0 \\ n & 1 & 2 & -3 & 0 & \cdots & 0 & 0 & 0 \\ n ...
2
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1answer
40 views

Determinant problem $n\times n$ with $0$'s and $1$'s

$$D_n=\left\vert\begin{matrix}0&1&0&0&\cdots&0&0\\ 1&0&1&0&\cdots&0&0\\ 0&1&0&1&\cdots&0&0\\ ...
2
votes
3answers
59 views

The Hadamard determinant problem: understanding a proof for the upper bound on matrix determinants

I'm working through a proof on The Hadamard determinant problem which can be found in Proofs from THE BOOK. I don't understand how the transition from real valued matrices $A$ with entries in ...