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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53 views

Why this is not always true $\det(A +B ) = \det(A) + \det(B)$ for square matrices [duplicate]

Why $\det(A +B ) = \det(A) + \det(B)$ is not always true for square matrices $A$ and $B$? Note that $A$ and $B$ are square matrices.
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1answer
30 views

Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
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1answer
14 views

Characteristic polynomial: Identity permutation?

This concerns the characteristic polynomial of a matrix. http://www.math.umn.edu/~olver/num_/lnv.pdf p. 7 (or p. 92). every term is prescribed by a permutation π of the rows of the matrix ...
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0answers
33 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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2answers
34 views

Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
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16 views

A determinant that arises when proving the Alternating Sign Matrix Conjecture

Prove that $$\det\bigg(\frac{1-s^{i+j-1}}{1-t^{i+j-1}}\bigg)^n_{i,j=1}=t^{n^3/3-n^2/2+n/6}\prod_{1\leq i<j\leq n}(1-t^{j-i})^2\prod_{i,j=1}^n\frac{1-st^{j-i}}{1-t^{i+j-1}}$$ In his book, D. ...
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2answers
34 views

If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalizable over ...
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64 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
0
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1answer
20 views

Determinant of block matrix with null row vector

I'm a bit confused on a problem. I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1x1) ...
1
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1answer
41 views

How does determinant expansion by different rows work?

I have almost always seen the determinant expanded by using the first row: $$ A = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} $$ Such as: $ |A| = ...
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5answers
93 views

Are there singular matrices such that if we change any entry it will be non-singular?

Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix with real entries such that its determinant is zero, but if one changes any single entry one gets a matrix with non-zero ...
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1answer
21 views

Changing the Form of this Factorisation

I'm brushing up on some high school maths and I'm currently revisiting determinants, specifically the factorisation of determinants. I'm working my way through a problem set and I keep getting stuck ...
2
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1answer
15 views

Finding char polynomial in $Z_3$

$ K=Z_3 $ $ A \in K_{(4 \times 4)} $ $$A= \begin{bmatrix} a & -1 & -2 & -2 \\ 0 & a-1 & -2 & 0 \\ -2 & 0 & a & 0 \\ -2 & -1 & 0 & a-2 \\ ...
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2answers
31 views

Brief moment in theorem about determinant from baby Rudin

If $(j_1,j_2,\dots, j_n)$ is an ordered $n$-tuple of integers, define $$s(j_1, j_2, \dots, j_n)=\prod \limits_{p<q}\text{sgn}(j_q-j_p).$$ Let $[A]$ be the matrix of a linear operator on ...
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1answer
24 views

How to prove that determinant can take any real value using only this definition of the determinant?

I was reading some facts about the determinant and refreshed my memory with the fact that the determinant of the $ n\times n $ matrix can be defined as $ \det(A)=\sum_{\sigma \in S_n} sgn(\sigma) ...
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1answer
49 views

Determinant of a matrix

I'm calculating the determinant of the matrix below. However the right answer is $-15$, but I'm getting $-30$. Can someone please point out the mistake?
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1answer
44 views

About a definition of “rank” of a matrix.

I am familiar with the definition of rank of a matrix as either (1) the maximal number of linearly independent rows or columns or (2) as the dimension of the image of the matrix. Another ...
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2answers
40 views

Sign of determinant of a $3 \times 3$ matrix with entries $1+\alpha^{i+j-2}+\beta^{i+j-2}$, for distinct $\alpha,\beta\in\mathbb R\setminus\{1\}$

Let $ \alpha\ne1,\beta\ne1$ be the distinct real roots of the equation $$ax^2+bx+c=0,~~a,b,c\in \mathbb{R},a\ne 0$$ Let $S_n=\alpha^n+\beta^n,n\geq0$ and ...
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2answers
19 views

Determinant property $|c \cdot A| =c^n \cdot |A|$

$$\begin{array}{|ccc|} x & 2 & 4 \\ x & 1 & 2 \\ x & 4 & 0 \\ \end{array} = x \cdot\begin{array}{|ccc|} 1 & 2 & 4 \\ 1 & 1 & 2 \\ 1 & 4 & 0 \\ ...
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2answers
18 views

Inverse and determinant of complex matrix

Is the determinant calculated the same way as a real matrix? Also when does $A^{-1}$ exist? Should the determinant be different from zero? a real number? or any complex number?
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1answer
50 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
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Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...
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76 views

Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
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3answers
94 views

Calculating determinant of 100x100 matrix

I was trying to calculate the determinant of 100x100 matrix: ...
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1answer
28 views

$4\times 4$ Matrix determinant (For computer graphics)

So Opengl and other graphics Api's use Matrices that are $4\times 4$, because they have to include affine transformations (translation). The 4th row and column are included for this reason. The ...
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2answers
38 views

How to prove that $ A^TA$ is singular for $2\times 3$ matrix $A$

I was trying to find the determinant for $A^TA$ where $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ \end{array} \right) $$ I tried out with some numbers in place of $a, ...
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2answers
29 views

Jordan form of the “multiplicative table” matrix

I have to find the Jordan form of the $(10\times10)-$matrix $A$ with the $n$th row formed by $n(1,2,3,4,5,6,7,8,9,10), \ \ 1 \leq n \leq 10$ I have calculated the determinant of $(A-xI)$ using ...
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0answers
31 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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62 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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23 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
43 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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49 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
27 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
32 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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64 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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23 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
21 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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27 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
82 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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48 views

Exact determinant of a circulant matrix

The wikipedia gives us a formula for the determinant of a circulant matrix. That is: $$\mathrm{det}(C) = \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \dots + c_1\omega_j^{n-1})= ...
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47 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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2answers
58 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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1answer
13 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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16 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
23 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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1answer
81 views

If $\det{(H_{p}^{\infty})}$ converges to a constant value, estimate the range of $p$.

Introduction: One day I calculated the value of determinant which is like Hilbert matrix $H_{p}^{n} \in \bf{R}^{\it{n \times n}}$using my computer. The determinant is defined below. $$ ...
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2answers
31 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 ...