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
2answers
96 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&a_1&\cdots&a^{n-1}_1\\ 1&a_2&\cdots&a^{n-1}_2\\ ...
7
votes
1answer
261 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). ...
0
votes
6answers
2k 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) ...
0
votes
2answers
192 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 ...
1
vote
0answers
86 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 ...
0
votes
2answers
102 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 ...
5
votes
2answers
465 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 ...
8
votes
2answers
149 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
votes
0answers
77 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 ...
1
vote
0answers
69 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) & ...
2
votes
2answers
226 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 ...
2
votes
1answer
72 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 ...
1
vote
2answers
582 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?
0
votes
3answers
70 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
votes
1answer
151 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?
2
votes
2answers
352 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
votes
3answers
784 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 ...
3
votes
1answer
68 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
votes
2answers
78 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 ...
10
votes
2answers
381 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 ...
0
votes
3answers
318 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| = ...
1
vote
1answer
2k 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 ...
1
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2answers
105 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 ...
1
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0answers
33 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 ...
2
votes
1answer
52 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
votes
1answer
439 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 ...
1
vote
1answer
105 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 ...
1
vote
1answer
65 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, ...
3
votes
1answer
60 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
votes
2answers
118 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 ...
3
votes
0answers
56 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 ...
0
votes
0answers
38 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 ...
1
vote
1answer
42 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 ...
2
votes
1answer
202 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 $|A| = -(a-b)(b-c)(c-a)(a+b+c)$.
3
votes
1answer
116 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 ...
8
votes
4answers
394 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
votes
1answer
49 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
votes
1answer
131 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
1
vote
1answer
44 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 ...
3
votes
2answers
109 views

A function that looks like determinant

Let $A$ be the $n\times n$ matrix $(a_{ij})$. By Laplace formula, the cofactor expansion along the $j$th row is $$\det(A)=\sum_{j=1}^n (-1)^{i+j}a_{ij}M_{ij}.$$ I'm studying the function ...
1
vote
1answer
174 views

Find determinant of the matrix NxN

We are given matrix $M_{n,n}$, where $m_{ij} = \begin{cases} a_i \cdot a_j,\ \mbox{if}\ i \ne j \\a_i^2+k,\ \mbox{if}\ i=j \end{cases}$ Hence, M gotta look like that: $ \left( \begin{array}_ ...
3
votes
3answers
83 views

Vandermonde determinant for order 4

I'd like to show the case $n=4$ for the Vandermonde-determinant. It should look like this: $V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & ...
2
votes
1answer
22 views

Proof x \in L \leftrightarrow det(…) = 0.

I just need some help with the following proof: Let $v = (v_1,v_2) $and $ w=(w_1,w_2)$ be two points in $K^2 , v \not= w$ and $L \subseteq K^2 $ a line through these two points. Show that ...
0
votes
2answers
46 views

Proving a Simple equation

I have a not so smart question; but I just cannot figure it out ! Suppose that I have a real $2 \times 2 $ matrix $(a_{ij})$ of non-zero determinant, and let $z \in \mathbb{C} $ be such that $ ...
4
votes
2answers
267 views

Why and When is a determinant of a larger matrix equal to a determinant of a smaller matrix?

The following is written in the solution of my textbook. $$|A|= \left| \begin{array} {cccc} 1 & 2& -1& 4 \\ 0& 5& -1& 6 \\ 0& -3& 3& -6 \\ 0& 2& 2& ...
3
votes
2answers
54 views

Determinant algebra

If $A$ and $B$ are $4 \times 4$ matrices with $\det(A) = −2$, $\det(B) = 3$, what is $\det(A+B)$? At first I approached the problem that $\det(A+B) = \det(A) + \det(B)$ but this general rule would ...
1
vote
1answer
63 views

Determinant Of Matrix (A) - Confusion about wording of the question.

Okay, So I'm a bit confused on what to do for this question. I figured out that Det(B) is just the determinant of matrix A and that matrix B is just the upper-triangular version of Matrix A. But how ...
3
votes
2answers
73 views

What is this math problem asking for?

I have a problem with a problem. I don't know how it is asking me to proceed, even though I know how to do it any which way. I just need to understand what the english means! Problem: Determine the ...
1
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1answer
332 views

Simple way to find the sign of a determinant given a singular value decomposition

Consider a quadratic $n\times n$ Matrix $A$ and the general question "how to find the determinant $\det(A)$ when too lazy for a Laplace Expansion but lucky enough to get a singular value decomposition ...
1
vote
1answer
105 views

Linear Algebra, meaning of 0 determinant in linear transformations

Lets say the area of a figure in $\Bbb R^2$ was $10$. Then after a noninvertible linear transformation from $\Bbb R^2$ to $\Bbb R^2$, is there enough info to determine the new area? Since its ...