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
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3answers
42 views

How to explain the calculation of the determinant of a $4\times4$ matrix

In my linear algebra lecture notes, I am studying an example which concerns the calculation of the determinant of a $4 \times 4$ matrix, by first reducing the matrix to upper triangular form. (See ...
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0answers
21 views

Determinant of block-triangular matrix made of 3 matrices [duplicate]

Let $A$ be a $k \times k$ matrix and $B$ be a $\left(n-k\right) \times \left(n-k\right)$ matrix, and $Z$ be the $n \times n$ matrix $$ Z = \left( \begin{matrix} A & C \\ 0 & B ...
1
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2answers
35 views

Determinants order of operations

When computing determinants using their properties, what is the order in which the determinant gets evaluated? Ie. \begin{vmatrix} 2AA^t \\ \end{vmatrix} Do we start with $2A$ or ...
2
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2answers
51 views

Computing the volume of a fundamental domain of a lattice

Suppose I have $n$ linearly independent vectors in $\mathbb{R}^m$, say $v_1, .., v_n$. Then $v_1,..., v_n$ form a lattice $\Lambda$ inside a subspace $V$ = $\mathbb{R}v_1 + ... + \mathbb{R}v_n ...
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1answer
36 views

Why is determinant called volume of the fundamental parallelepiped in geometry of numbers?

Let $v_1, ..., v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$. Then they form a lattice $\Lambda \subseteq \mathbb{R}^n$ and the volume of the fundamental domain is $|\det A|$, where $A$ ...
1
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1answer
70 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
0
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3answers
34 views

Find determinant value

\begin{vmatrix} 3 & 2 & 0 & 0 & . &. & . & . &0 &0 \\ 1 & 3 & 2 & 0 & . &. & . & . &0 &0 \\ 0 & 1 & 3 & 2 & . ...
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0answers
30 views

The discriminant of (1,x,x^2) in a cubic field.

Let $K$ be a cubic field such that $K=\mathbb Q[x]$ with $x^3=2$. The discriminant of $\{1,x,x^2\}$ is supposed to be $\begin{vmatrix} 3 & 0 & 0 \\ 0 & 0 & 6 \\ 0 & 6 & ...
3
votes
3answers
124 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
0
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2answers
60 views

Same roots, same polynomial? How to prove characteristic polynomial of $AB = BA$?

I'm giving a (simple) proof that the characteristic polynomial of $AB$ = characteristic polynomial of $BA$ (without using the fact that $AB$ and $BA$ are similar). $det(AB) = det(A)det(B) = ...
5
votes
3answers
90 views

Prove that $\det(AA^T+I)\ge 1$

If $A$ is a matrix with real entries, prove that $$\det(AA^T+I)\ge 1.$$ I tried using the eigenvalues. One thing came into my mind: maybe $AA^T$ is positive definite (I don't know whether this is ...
5
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2answers
118 views

Prove: $(\det(A-B)+\det(A+B) )^2 \ge 4\det(A^2-B^2 )$

Let $A,B \in \mathcal{M}_n (\mathbb{R})$ two matrices so that: a) $AB^2=B^2 A$ and $BA^2=A^2 B$ b) $\text{rank}(AB-BA)=1$. Prove: $$(\det(A-B)+\det(A+B) )^2≥4\det(A^2-B^2 )$$ ...
0
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0answers
21 views

How do we sketch the ellipse determined by $T(\vec{x})$ and determine its axes, given an expansion factor?

I have been told that if $\left\{\exists \, T(\vec{x})^{-1}\mid T(\vec{x})=A\vec{x} \mid \mathbb{R}^2\mapsto\mathbb{R^2}\right\}$, then the image $T(\Omega)$ of the unit circle $\Omega$ is an ellipse. ...
2
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4answers
84 views

An Eigen Value of $\tiny \begin{pmatrix} a&b&1 \\ c&d&1 \\ 1&-1&0\\ \end{pmatrix}$ is :

Let $a,b,c,d$ be distinct non zero real numbers with $a+b=c+d.$ Then, an eigen value of the matrix $A= \begin{pmatrix} a&b&1 \\ c&d&1 \\ 1&-1&0\\ \end{pmatrix}$ is : $(i)~a+c ...
2
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1answer
75 views

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are ...
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2answers
64 views

Is the area of $T(\Omega)=|\det A|\,(\text{area of }\Omega)$?

We are given that $\Omega$ is a parallelogram in $\mathbb{R}^3$ and $\left\{ T(\vec{x}) = A\vec{x} \mid \mathbb{R}^3 \mapsto \mathbb{R}^3\right\}$ is a linear transformation. From the definition of ...
2
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1answer
77 views

What is the 3-volume of the 3-parallelepiped defined by $\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}$?

We have $\left\{\vec{v_1},\vec{v_2},\vec{v_3}\right\}=\left\{\begin{bmatrix}1\\0\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\2\\3\\4\end{bmatrix}\right\}$ ...
0
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1answer
55 views

can we use saras method for finding determinant of matrix greater than 3x3

Can Sarrus method of finding determinant be used for finding determinant of matrices greater than $3\times 3$ http://en.wikipedia.org/wiki/Rule_of_Sarrus as I am unable to find any example of matrix ...
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0answers
44 views

Induction proof: $\det(M) = \prod_{1 \le j \le n} (x_j - x_i)$

Following problem: Let $\mathbb{K}$ be a Field and $M = \begin{pmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ 1 & x_n & ...
0
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2answers
47 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...
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10 views

fix a line for operation on matrices

operations on matricies determinant Why does a line at least must be fixed to to operations on matrices for their determinant calculation? I understand that if it was allowed not to do such a ...
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0answers
31 views

$j$-volume of $j$ dimensional parallelepiped inside $\mathbb{R}^n$

Let $v_1, ..., v_j \in \mathbb{R}^n$ be linearly independent. Let $V = \mathbb{R}v_1 + ... + \mathbb{R} v_j$ be a subspace of $\mathbb{R}^n$ and $\Gamma = \mathbb{Z}v_1 + ... + \mathbb{Z} v_j$ a ...
3
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3answers
45 views

Determinant of the sum of some special matrix

$A,B$ are $3\times 3$ matrices. It is known that: $\det(A)=0$ $\forall i,j: b_{ij}=1$, where $b_{ij}$ is an element of matrix $B$ $\det(A+B)=1$ Find $\det(A+2014B)$ I don't know what to do. I ...
0
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1answer
67 views

Matrix derivatives of determinant and inverse related to $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$

I would like to calculate the derivatives of determinant and inverse related to the term $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$ with respect to $\mathbf{X}$, where $\mathbf{C}$ is a constant matrix. ...
1
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1answer
52 views

is it possible to use induction to prove the following?

I know for sure that there is some easy way to prove what I am about to tell, but, at first, I'd like to know if I can set up a proof by induction for two "cross-referenced formulas". I have two ...
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3answers
50 views

How do you prove (or disprove) the statement: If $A^3 = 0$, then A-I is non-singular

I've proven something similar: A*A =0, then A + I is non-singular for 2x2 matrices. But not sure how to proceed for $A^3 = 0$, then A-I is non-singular Also, not sure how to prove A*A =0, then A + I ...
3
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1answer
88 views

Inquiry about determinant of $ \left(\begin{matrix} A & B \\ B^T & C \end{matrix}\right)$

Based off of http://en.wikipedia.org/wiki/Determinant#Block_matrices, I'm trying to find the formula for $\det(M)$ when $M = \left(\begin{matrix} A & B \\ B^T & C \end{matrix}\right)$. It is ...
0
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3answers
38 views

Counter example to the converse of the special property of triangular matrices

Let $A = (a_{ij})$ be a square matrix of order $n$ over $\mathbb R$. Give a counter-example of: $det(A) = \prod_{t = 1}^n a_{tt}$ $\implies$ $A$ is a triangular matrix. Any help? Thanks.
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0answers
44 views

Area of parallelogram from a linear operator

let $L: \mathbb R^2\to\mathbb R^2$ be a linear operator which is invertible. Let {$u, v$} be a linearly independent set in $\mathbb R^2$. Find a formula for the area of the parallelogram induced by ...
0
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1answer
45 views

Using determinants to find a unique solution

I am supposed to find the values of $k$ for which the following system has a unique solution: $$kx+y+z=1\\x+ky+z=1\\x+y+kz=1.$$ I came up with this: ...
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1answer
34 views

Given a Percentage, Find the Smallest Integer Dividend for an Unknown Number of Integer Tests

In a classroom exam where each question is worth 1 point and the number of exam questions are unknown. If the student receives results of a test in a percentage with decimals, let's say ...
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2answers
23 views

Can we use $R_1 \to R_2-R_1$ as an elementary row operation without changing the value of the determinant?

Can someone help me out? I need to know if $R_1 \to R_2-R_1$ is a valid elementary row operation that can be used on the given determinant without changing the determinant's value. It's a $3\times 3$ ...
1
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0answers
65 views

Can we interchange one row and one column in a determinant?

Can we swap the ith row and the ith coloumn in a determinant as an elementary operation? What happens when we do interchange them? Does the value of the determinant remain constant or does it change?
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1answer
61 views

Find The Determinant Of A Finite Field Matrix

Let there be $A_n=\left(\begin{matrix}4&2&\cdots&2\\2&4&\ddots&\vdots\\\vdots&\ddots&\ddots&2 \\2&\cdots&2&4\end{matrix}\right)\in ...
4
votes
1answer
95 views

Invertible block matrix

Could I find $A_1,A_2,A_3,A_4 \in M_n(\mathbb C)$, such that, for all $z_1,z_2,z_3,z_4\in \mathbb C$, $\det(z_1A_1+z_2A_2+z_3A_3+z_4A_4)=0$ and $\det \begin{pmatrix} ...
0
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0answers
28 views

Proving properties about cofactors when matrix is not invertible

(1) $cof(A^t) = cof(A)^t$ (2) $cof(A)^t = det(A)I$ I have (at least, I think so) proofs of (1) and (2). But the proofs require the matrix $A$ to be non-singular. How do I prove (1) and (2) if $A$ is ...
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0answers
18 views

Deducing a Recursion formula for Vandermonde Matrix

Vandermonde matrix, $V_n(a_1, \dots, a_n)$ = $\left|\begin{array}{cccc}1 & a_1 & a^2_1 & ... & a^{n-1}_1 \\... & ... & ... & ... \\1 & a_n & a_n^2 & ... & ...
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3answers
40 views

Confusion about how the determinant changes when all rows are multiplied by a scalar

I am having some trouble thinking about properties of the determinant. I understand why it is true that if $B$ is the matrix obtained from an $n \times n$ matrix $A$ by multiplying a row by a scalar ...
0
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3answers
29 views

How do you solve a determinant thats set to a value with 1 unknown variable?

I was just wondering if someone could explain the steps you take to solve a determinant that has an unknown variable, and is set to equal integer value? For example: How is one supposed to go about ...
0
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2answers
44 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
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0answers
87 views

On the determinant of a certain matrix over the polynomial ring of $n$ variables over a field

Let $A = k[x_1,\dots, x_n]$ be a polynomial ring over a field $k$. Let $\sigma_1,\dots,\sigma_n$ be distinct permutations of the set $\{1,\dots,n\}$. Is the determinant det$(x_{\sigma_i(j)})$ ...
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1answer
30 views

Determinants and row operations

So multiplying a row by a constant multiple the determinant by the same constant and swapping 2 rows will multiply the determinant by a negative right and adding or subtracting does not change the ...
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0answers
13 views

finding determinant using elementary matrices

use elementary matrices to find the determinant of the matrix A A = (2 5 -1),(-1 -1 5), (3 7 -3) solution I found: -36
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1answer
28 views

Prove the determinant of A equals 2 times the determinant of B

Question: Prove that $$ det \begin{bmatrix} a + b & p + q & u + v \\ b + c & q + r & v + w \\ c + a & r + p & w + u \\ \end{bmatrix} ...
3
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1answer
410 views

Matrix with integer entries

The determinant of a given square matrix $A$, with rational entries, equals 1. It is known that all entries of $A^{2015}$ are integers. Is it true that all entries of $A$ are integers? My attemt: ...
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17 views

Showing a simple determinant property

I want to use Laplace expansion to show/prove to myself formally that the rule for determinants stating that if B is a matrix obtained from A , where $A \in \mathbb M_{nxn}$, by multiplying a row or ...
0
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0answers
100 views

Find Determinant

How can I find the determinant of a following matrix in the simplest way ? $$\begin{vmatrix} 1 & 10 & 100 & 1000 & 10000 & 100000 \\ 0,1 & 2 & 30 & 400 & 5000 ...
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1answer
40 views

Argument that Vandermonde matrix's determinant has $n-1$ distinct roots

det(Vandermonde) = $\left|\begin{array}{ccccc}1 & x & x^2 & ... & x^{n-1} \\1 & a_2 & a^{2}_{2} & ... & a^{n-1}_{2} \\1 & ... & ... & ... & ... \\1 ...
1
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1answer
41 views

Finding the eigenvalues of a matrix problem

So I do know how to compute the eigenvalues of a matrix. At least, that's what I thought. I got the matrix A = \begin{bmatrix}1&-2&0\\-2&0&2\\0&2&-1\end{bmatrix} My approach ...
1
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1answer
40 views

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ ...