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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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
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2answers
361 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
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2answers
134 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
76 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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0answers
64 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) & ...
1
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2answers
179 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 ...
1
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1answer
63 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
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2answers
491 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
60 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
146 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
262 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
595 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
50 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
76 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
340 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
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3answers
260 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
1k 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
95 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
30 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
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
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1answer
394 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
82 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
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
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1answer
58 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
108 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
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0answers
55 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
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0answers
31 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
40 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
154 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
43 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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0answers
42 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 ...
3
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1answer
115 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
379 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
48 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
118 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
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1answer
41 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
106 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 ...
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1answer
150 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
74 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 ...
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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
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2answers
261 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& ...
2
votes
2answers
52 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 ...
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1answer
61 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
260 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 ...
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1answer
92 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 ...
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3answers
43 views

Linear Algebra, find determinant with x1, x2,…,xn as scalars

I have no clue how to even begin solving for $\det(A)$ since $n$ is unknown, HELP!
0
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2answers
30 views

finding the rank of following matrix, please check it

\begin{pmatrix}3&0&1&2\\4&7&3&3\\1&7&2&1\end{pmatrix} please find its rank, I got the answer 3, is it correct? please check it
0
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3answers
66 views

Finding inverse of matrix

Find the inverse of the following matrix$$\begin{pmatrix}ab&0\\0&1\end{pmatrix}$$ I found $$\begin{pmatrix}\frac{1}{ab}&0\\0&1\end{pmatrix}$$ but one of my friend got ...