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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43 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
6
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2answers
199 views

Determining the derivation of a determinant

Let $\Phi\colon E\to M$ with $E\subset \mathbb{R}\times M$ and $M\subset\mathbb{R}^n$ open. Consider the function given by $x\mapsto \Phi(t,x)$ for fixed $t\in\mathbb{R}$. (1) Determine $$ ...
0
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1answer
25 views

Proof regarding effect of row operations on determinants>

Let $A,B \in K^{n,n}$ and suppose $B$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$. Prove $det(A)=det(B)$. My Attempt I tried to use proof by induction for this . Take ...
1
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1answer
37 views

Find determinant $\det M$, where $m_{ij}=a_ia_j$, and $m_{ii}=a^2_i+k$

Let ${a_1,\dots,a_n}$ --- sequence and $k\ne 0$. Define matrix $M$ in following way: $m_{ij}=a_ia_j$ if $i\ne j$, and $m_{ii}=a^2_i+k$. Find $\det M$.
2
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1answer
60 views

Coordinate-free proof of determinant of transpose

I'm interested in a coordinate-free proof of the statement $\mathrm{det}(A) = \mathrm{det}(A^T).$ Let $V$ be a finite-dimensional vector space over a field $K$, and let $f : V \rightarrow V$ be an ...
7
votes
2answers
108 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
2
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1answer
53 views

If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
0
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2answers
125 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
12
votes
1answer
162 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
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4answers
60 views

Find matrix determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...
2
votes
3answers
64 views

Is there any way to check wheter the determinant of a matrix $A$ with $|\text{det }A|=1$ is positive or negative?

Let $A\in\text{GL}(n,\mathbb{R})$ with $|\text{det }A|=1$. Is there any way to check wheter $\text{det }A$ is positive or negative without computing it?
0
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0answers
33 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
0
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0answers
23 views

Representative value of non-square matrix

First of all, I apologise if this question is inappropriate, I wish I could be more specific - but due to the nature of it, as I am actually asking for a suggestion of some technique, that's hard to ...
12
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3answers
204 views

determinant inequality $\det(A^2+AB+B^2)\geq\det(AB-BA)$

$A,B$ are two $2\times 2$ real matrices, then $$\det(A^2+AB+B^2)\geq\det(AB-BA)$$ The inequality is equivalent to the following problem: Let $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
0
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2answers
70 views

Odd-dimensional skew-symmetric matrix is singular, even in a field of characteristic 2

I'm familiar with the usual proof $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A)$ which only works in fields of characteristic not equal to 2. To get a proof that works in characteristic 2 I can ...
0
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1answer
26 views

Determinant of a matrix over a field K

Let $A$ be an $n \times n$ matrix over a field $K$. Do the properties of the determinant of a real matrix hold for the matrix $A$? If not, in which fields do the properties of the determinant of a ...
0
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2answers
62 views

finding determinant for matrix using upper triangle method!

so Here an example for matrix that I'm trying to evaluate its determinant! | 1 3 2 1| | 0 1 4 -4| | 2 5 -2 9| | 3 7 0 1| when applying first row operation i get | 1 3 2 1| | 0 1 ...
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1answer
22 views

meanings determinants of matrices in finite field

Let's $\Bbb{Z}_q$ is finite field. ($q$ is prime number). Lets $A_1$ – set of matrices $n\times n$, such that $\det(M) = 1$, for any matrix $M \in A_1, A_2$ – set of matrices $n\times n$, such that ...
3
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2answers
57 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
3
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0answers
27 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
1
vote
2answers
45 views

Find $2\det ( \frac{1}{2} A )$ given that $A$ is $3\times 3$ and $\det(A)= -2$

Here is a question that should be done today: If $A$ is $3\times 3$ and $\det(A)= -2$, find $2\det(\frac{1}{2}A)$. I solved this problem but I am not sure because the way I used is not accurate! ...
7
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4answers
280 views

Determinant of a Special Symmetric Matrix

If $A$ is a symmatric matrix of odd order with integer entries and the diagonal entries $0$ then $A$ has determinant value even. I can prove the result if I can show that the eigenvalues of $A$ are ...
2
votes
4answers
112 views

How to prove the inequality $\det (AA^T) \ge 0$?

How to proof for any matrix $A \in \Bbb R^{n \mathbf x n}$, that the next inequality $\det(AA^T) \ge 0$ is true?
3
votes
3answers
122 views

Determining the values of $\lambda$ for which the matrix is invertible

I'm working on a homework problem and am a little stuck. The question is: Determine the values of $\lambda$ for which the matrix $$\begin{pmatrix} \lambda &-1&0\\ -1&\lambda&-1\\ ...
1
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2answers
296 views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
6
votes
5answers
405 views

Find the determinant of the following;

Find the determinant of the following matrix, and for which value of $x$ is it invertible; Now I don't really know how to procees as I get find a suitable row operations that will simplify ...
0
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1answer
27 views

volume of parallelotopes

I know that determinant indicates the volume of a parallelotopes spanned by the n vectors. I absolutely understand that the properties of a determinant: any function $f:\mathbb{R}^{n\times ...
0
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0answers
33 views

How to prove this identity? (About determinant)

How to prove the determinant of \begin{equation} \left( \begin{array}{cccccc} a_{00} & a_{01}& a_{02} & \cdots & \cdots& a_{0k}\\ 1 & a_{11}& a_{12} & \cdots ...
0
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1answer
26 views

Find $det(xy^T)$ where $x$, $y$ are vectors from $R^n$, $n$>1 [duplicate]

I represented $x$ as $[x_1\ x_2\ ... x_n]^T$, and $y^T$ as $[y_1\ y_2\ ... y_n]$. Multiplying them produces a matrix $n$x$n$: $$ \begin{pmatrix}x_1y_1&x_1y_2&\dots& x_1y_n\\ ...
3
votes
1answer
47 views

Prove positive definiteness

I want to prove that the matrix $$\begin{pmatrix} 1 &\cfrac{1}{2} &\cfrac{1}{3} &\cdots &\cfrac{1}{n} \\ \cfrac{1}{2} &\cfrac{1}{3} &\cfrac{1}{4} &\cdots ...
0
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0answers
49 views

Help proving by induction that the determinant is equal to: $(-1)^{\frac{n(n-1)}{2}}\cdot a_{1n}\cdot a_{2n-1} \cdots a_{2n1}$

Help proving that the determinant is equal to $(-1)^{\dfrac{n(n-1)}{2}}\cdot a_{1n}\cdot a_{2n-1} \cdots a_{2n1}$ $$ \begin{vmatrix} 0 &0 & \dots &0 &a_{1n}\\ 0 &0 & \dots ...
2
votes
1answer
18 views

Computing determinants using derivatives in an arbitrary field

When computing determinants that depend on a parameter $t\in \Bbb R$, it is often useful to use the fact that \begin{align} \det(V_1(t),\dots,V_n(t))&=\det(V_1(a),\dots,V_n(a))+\\ ...
2
votes
2answers
51 views

real matrix with $Tr((A-I)^{T}(A-I) )<1$

$A$ is a $n\times n$ real matrix, $$\operatorname{Tr}((A-I)^{T}(A-I) )<1$$ then $\det(A)\ne0$. well, $$\sum_{i\ne j}a_{ij}^2+\sum (1-a_{ii})^2\lt1$$ How to derived $\det(A)\ne0$? Thank ...
1
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0answers
53 views

probability of having a non-zero determinant

$K=\mathbb{Z}_p$ for some prime p, and $dim V = n$. It has been shown that the number of different bases in $V$ is: $\frac{1}{n!} \prod_{i=0}^{n-1}(p^n - p^i)$ (bases which are permutations of one ...
-1
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1answer
53 views

How to prove that $\det\left[\pmatrix{u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3}\pmatrix{s_1 & s_2 & s_3\\ t_1 & t_2 & t_3}\right]=0$?

Evaluate $\det\left[\begin{pmatrix} u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3 \end{pmatrix} \begin{pmatrix} s_1 & s_2 & s_3\\ t_1 & t_2 & t_3 \end{pmatrix}\right]$. I ...
2
votes
1answer
33 views

Prove that determinant of matrix equal to n

Prove that determinant of matrix $D_n$ (square $n$ x $n$ matrix) is equal to $n$. $$ \begin{matrix} 1 & -1 & -1 & \cdots & -1 \\ 1 & 1 & & & \\ 1 & & 1 & ...
3
votes
0answers
29 views

Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
1
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0answers
44 views

Prove that every proper principal submatrix of $\lambda I-A$ is nonsingular under certain assumptions

Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda ...
0
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2answers
87 views

Determinants Problem [closed]

\begin{align}\begin{vmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2\end{vmatrix} = 2abc(a+b+c)^3\end{align} Determinant proof ...
2
votes
3answers
93 views

maximum value of $\det(A)$, elements $0, 1, 2, 3$,

$A$ is a $3\times 3$ real matrix, whose elements can be $0, 1, 2, 3$. What is the maximum value of $\det(A)$? $\det(3I)=27$, the maximum value should be $\gt27$. Thank you very much for your ...
5
votes
1answer
114 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
7
votes
2answers
284 views

Circulant determinants

Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true? There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the ...
3
votes
1answer
124 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
0
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0answers
28 views

An endomorphism sending a basis element to zero

Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$. Let $u$ be the endomorphism $$u(P)=(X^2-1)P''-2XP'$$ I want to determine the determinant of $u$. So I proceed by ...
0
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1answer
56 views

$k$-dimensional volume of the simplex spanned by $(k+1)$ vectors in $\mathbb{R}^n$ for $k<n$

My question is about the $k$-dimensional volume of the simplex spanned by the origin together with $k$ vectors stored in an $k \times d$-matrix A. I found two references saying that this volume is ...
8
votes
4answers
929 views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
3
votes
1answer
67 views

What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
3
votes
1answer
68 views

Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
4
votes
1answer
185 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
0
votes
0answers
30 views

How can I compute pseudo determinant

Let A square n by n matrix and let b:=pseudo det of A And assume that A is diagonalizable and rkA=r Then what is pseudo det of AA^(t)??