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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1answer
38 views

Exploring Determinants of Matrices. [closed]

I have a homework and i have to explore different patterns of determinant. I have find a unique pattern with determinants and make a conjecture. Your ideas about different patterns will be welcomed. ...
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1answer
21 views

Proof, wheather a subset of a Group is a Subgroup

I have to check, weather the following subset of a group is also a subgroup: $$U = \left\{ \begin{pmatrix} a & -b \\ \overline{b} & \overline{a} \end{pmatrix} \in GL(2, \mathbb{C}) \bigg\vert ...
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1answer
44 views

Proof of Nonnegative Determinant

How can I solve this question?
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1answer
34 views

Which is the max value of the determinant with 4 lines and 4 colums ,where every term is +- 1?

I understand that this problem can be solve with the volume of a tetrahedron. But i don't know how. please help me ! \begin{vmatrix} \pm1 & \pm1 & \pm1 & \pm1 \\ \pm1 & \pm1 & ...
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1answer
40 views

Formula for determinant of this matrix

Let's have matrix $(n-1) \times (n-1)$ $$ \begin{pmatrix} 3 & 1& 1& \cdots& 1 \\ 1 & 4& 1& \cdots& 1 \\ 1 & 1& 5& \cdots& 1 \\ \vdots &\vdots ...
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0answers
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Let A and B be matrices with same dimension. Prove $|\det({}^tA\times B)|^2\leq\det({}^tA\times A)\cdot \det({}^tB\times B)$ [duplicate]

Let $A$ and $B$ be matrices of the same dimension. Prove $|\det({}^tA\times B)|^2\leq\det({}^tA\times A)\cdot \det({}^tB\times B)$, where ${}^tA$ is the transpose of matrix $A$ and $\det$ is the ...
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0answers
38 views

Proof of a property of a cofactor matrix.

If $A$ is a matrix with $n\geq2$, prove the following property of its cofactor matrix - $ {cof} (A^t) = ({cof} (A))^t$. Are the following properties of matrices and determinants of use here - (a) $ ...
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2answers
94 views

Prove Derivative is sum of determinants

Given $n^2$ functions $f_{ij}$, each differentiable on an interval (a,b), define $F(x) = det[f_{ij}(x)]$ for each $x$ in $(a,b)$. Prove that the derivative $F'(x)$ is the sum of the determinants, $$ ...
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0answers
46 views

Calculate determinant of Vandermonde using specified steps.

$V_n(a_1,a_2\dots, a_n)$ is a $N\times N$ Vandermonde matrix = $$\left|\begin{array}[cccc] 11&z_1&\cdots&z^{n-1}_1\\ 1&z_2&\cdots&z^{n-1}_2\\ ...
7
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1answer
247 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
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5answers
278 views

Why does a matrix have determinant zero if one row is the sum of two other rows?

So basically here I am trying to understand why it is like that? Suppose Matrix $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ a+d & b+e & c+f \end{array} \right) ...
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2answers
113 views

Determinant of block $n \times n$ matrix

Let det $A = \det(\begin{bmatrix}B& 0\\ 0& I_mI\end{bmatrix})$; $B$ and $D$ are square matrices. $I_m$ is an identity matrix of size $m$. I keep reading that it is obvious that we can view ...
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0answers
35 views

Cramer's rule and understanding Area/Volume

I'm having trouble connecting all the ideas we're learning in Linear Algebra. On the one hand, I understand how to find determinants, and therefore expansion factors. I also am fairly certain I have a ...
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2answers
60 views

Linear Algebra Review Questions

So I have a test on Monday and my professor posted a couple of non-graded review questions that she said we should look over. Anyhow, I have a couple of questions that I'd like answered if that's ...
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0answers
39 views

Generalisation of Gramian determinant

i'm wondering about those facts of basic linear algebra: if you have $n$ vectors $x_1,...,x_n \in \mathbb{R}^n$, you can easily test their linear dependance by computing their Gramian Matrix $M$ whose ...
4
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2answers
120 views

Determinant of rank-one perturbation of a diagonal matrix

Let $A$ be a rank-one perturbation of a diagonal matrix, i. e. $A = D + s^T s$, where $D = \DeclareMathOperator{diag}{diag} \diag\{\lambda_1,\ldots,\lambda_n\}$, $s = [s_1,\ldots,s_n] \neq 0$. Is ...
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2answers
111 views

Matrix with determinant 0

If $A \in M_3(\mathbb{R})$ is a $3 \times 3$ matrix with $\det(A)=0$ and the square of each element equals its cofactor, do we necessarily have $A=0_3$? $a_{ij}^2=A_{ij}$, where ...
3
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0answers
68 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
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53 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
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2answers
75 views

Is determinant of matrix multiplied its transpose always positive?

Assume $A$ is an arbitrary $m\times n$ real matrix. Is $\det(AA^T)$ always positive? Is it non-negative or it can have any value? Edit: It seems I have to emphasis that $m \ne n$ i.e. matrix is ...
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1answer
40 views

Can a tridiagonal matrix be rectangular?

My program works with tridiagonal matrices (calculates its LU decomposition) so before doing anythig, it stores the matrix in 3 vectors: the three diagonals only. So far my conclusion was, a ...
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2answers
111 views

The determinants of upper triangular matrices (For any 2x2 and 3x3 matrix)

I am trying hard to figure out what am I supposed to do, if I am supposed to go on write a conjecture about the particular question. How can I go on about to prove it?
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3answers
49 views

Determinants and eigenvectors [duplicate]

Hello, I'm trying to work through this question. I define linearly independent as: $a_1*v_1+a_2*v_2+...+a_n*v_n = 0$ iff every $a_i=0$. I also know that an eigenvector is a vector $v$ such that: ...
3
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1answer
130 views

A determinant inequality

Let $A,B$ be two $m\times n$ real matrices. Then $$|AA'|\cdot |BB'|\geq |AB'|^2.$$ For square matrices, it is the equality. How to prove this inequality then?
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2answers
70 views

Row swap changing sign of determinant

I was wondering if someone could help me clarify something regarding the effect of swapping two rows on the sign of the determinant. I know that if $A$ is an $n\times n$ matrix and $B$ is an $n\times ...
2
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3answers
268 views

Find the determinant without row expansion

Show that the determinant of the matrix \begin{bmatrix} 1& a& a^3\\ 1& b& b^3\\ 1& c& c^3\end{bmatrix} is $(a-b)(b-c)(c-a)(a+b+c)$ without expanding. I was able to get out ...
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0answers
41 views

What makes the permanent lot more difficult than the determinant

The permanent of an $n$-by-$n$ matrix $A$ = $(a_{i,j})$ is defined as: $\operatorname{perm}(A)=\sum\limits_{\sigma\in S_n}\prod\limits_{i=1}^n a_{i,\sigma(i)}$. ...
2
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2answers
65 views

Prove that $\det(A)=\det(A^T)$ algebraically

If we use row operations to turn matrix $A$ into an upper triangular matrix then the $\det(A)$ is equal to the product of the entries on its main diagonal. So if we transpose $A$, then those row ...
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2answers
150 views

The “second derivative test” for $f(x,y)$

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
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3answers
120 views

Proof using properties of determinants

I have to use properties of determinants to show that $$\left| \begin{array}{ccc} b^2+c^2 & ab & ac \\ ab & a^2+c^2 & bc \\ ac & bc & a^2+b^2 \end{array} \right| = ...
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1answer
112 views

Prove: If A is invertible, then adj(A) is invertible and $[adj(A)]^{-1}=\frac{1}{det(A)}A=adj(A^{-1})$

I can show the left side: $$A^{-1}=\frac{1}{det(A)}adj(A)$$ $$AA^{-1}=\frac{1}{det(A)}A*adj(A)\longrightarrow I=\frac{1}{det(A)}A*adj(A)$$ and, $$A^{-1}A=adj(A)\frac{1}{det(A)}A \longrightarrow ...
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2answers
63 views

Factorizing Determinants

I don't know how to factorize the determinants. Please help. 1. $$ \begin{vmatrix} a+b &b+c &c+a\\ b+c &c+a &a+b\\ c+a &a+b &b+c \end{vmatrix} $$ 2. $$ \begin{vmatrix} a^2 ...
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0answers
17 views

Preservation of determinants mod some ideal

Given a matrix with entries drawn from some field or commutative ring, what are the conditions for the determinant to be preserved mod some ideal? For a concrete example, I am thinking of matrices ...
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1answer
42 views

Prove $f$ not continuous at SEEMOUS Contest

Let $n$ be a nonzero natural number and $f:\mathbb{R}\to\mathbb{R}\setminus\{0\}$ be a function such that $f(2014) = 1 − f(2013)$. Let $x_1,x_2,x_3,...,x_n$ be real numbers not equal to each other. ...
4
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1answer
313 views

Cramer's Rule Question

Use Cramer's rule to solve this system for z: $$2x+y+z=1$$ $$3x+z=4$$ $$x-y-z=2$$ so my work is: $$\frac{\left|\begin{matrix} 2 & 1 & 1\\ 3 & 0 & 4\\ 1 & -1 & 2 ...
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1answer
43 views

Determinant by applying Gaussian Elimination

I understand when using Gaussian Elimination you have to get it in ref form (upper triangle) and calculate the product of the diagonal. Additionally you have to keep track of the number of swaps to ...
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1answer
63 views

Solve for $x$ in the given determinant.

Solve for $x$. $$ \begin{vmatrix} x^2-a^2&x^2-b^2&x^2-c^2\\ (x-a)^3&(x-b)^3&(x-c)^3\\ (x+a)^3&(x+b)^3&(x+c)^3\\ \end{vmatrix}=0. $$ I could factorise each term, ...
2
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1answer
45 views

$\mathrm{GL}_n$-representation theory question or a Tale of Two Determinants

The irreducible representations of $\mathrm{GL}_n(\mathbb C)$ are indexed by partitions $\lambda$. These representations are denoted by $\mathbb S_{\lambda}(V)$, where $V$ is the standard ...
2
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2answers
71 views

Geometric interpretation of determinant

I am trying to prove geometrically, without invoking the dot or cross products or orthogonality, that the volume of a parallelepiped formed by vectors $ \begin{bmatrix} a_1 \\ a_2 \\ a_3 ...
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0answers
52 views

Interesting determinant problem [duplicate]

how to go about computing following determinant? I tried using Gaussian elimination on some special cases and figured there might be some pattern, maybe a recurrence relation involved, but I just ...
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0answers
22 views

Basis and their orientation

let V be a vectorspace with $v_1 = (3,2,1), v_2 = (2,2,1), v_3 = (1,1,1)$. Do the two basis $A = (v_1, v_2, v_3)$ and $B = (v_2, v_3, v_1)$ have the same orientation? Since this is a new thematic for ...
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1answer
33 views

Proof cofactor-matrix cofac(AB) = cofac(B)*cofac(A)

Let $A \in K^{nxn}$ and $Cofac(A)$ be the cofactormatrix to A. I have to show (1) $cofac(AB) = cofac(B)*cofac(A)$. In fact I have: $^t(cofac A) = cofac (^t A) = adj(A).$ Then I have (I have ...
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1answer
61 views

Matrices and determinants question.

Establish that if A is the matrix \begin{bmatrix} b+c & a^2 & a \\ c+a & b^2 & b \\ a+b & c^2 & c \\ \end{bmatrix} then $|A| = -(a-b)(b-c)(c-a)(a+b+c)$.
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0answers
34 views

Quadratic matrix = regular matrix * diagonal matrix?

Is the following true? Let $A$ be an $n\times n$ matrix with $\det A\neq0$. Then there exists a regular matrix $S$ and a diagonal matrix $D(1,\dotsc,1,\det A)$ with $A=SD$. Some examples I made ...
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40 views

Determinant of a transformation matrix

I have been reading about determinants and transformation matrices. After that I was reviewing some exercises on a book I got. In one exercise I'm asked to find the transformation matrix and the ...
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1answer
93 views

Calculating the determinant gives $(a^2+b^2+c^2+d^2)^2$?

I need to calculate the following determinant in order to prove the following equality: $$\det\begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b ...
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4answers
310 views

What does it mean if $\det(A)$ equals $1$?

What does it mean if $\det(A)$ equals $1$? Does it mean that the identity matrix can be obtained from $A$ by only adding multiples of rows onto others?
2
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1answer
45 views

Compute the determinant-like sum

Let $A = (a_{ij} \mid i,j = 1, \ldots, 2n)$ be a skew-symmetric matrix. I want to compute the following sum: $$ S = \sum\limits_{\sigma \in S_{2n}} \mathop{\mathrm{sgn}}(\sigma)\, ...
2
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1answer
63 views

Trace and determinant of composition of a left-multiplication and a right-multiplication on a space of matrices

Determine the trace and determinant of the linear operator (on the space $\mathbb{F^{n\times n}}$) that sends the matrix $M\to AMB$ where $A$ and $B$ are $n\times n$ matricies
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1answer
35 views

Matrix determinant operations.

Suppose you are trying to find the determinant of the following matrix using the "upper triangulation" method: $\begin{matrix} 1&0&0\\ 0&1&0\\ 1&1&1 \end{matrix}$ If I take ...